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The Higher Mental Processes in 

Learning 



\S BY 

JOHN C: PETERSON, Ph.D. 

Professor of Psychology, State Agricultural College, Manhattan, Kansas 



A DISSERTATION 

Submitted to the Faculty 

OF the Graduate School of Arts and Literature 

In Candidacy for the Degree of 

Doctor of Philosophy 

Department of Psychology 



Private Edition, Distributed by 
The University of Chicago Libraries 
CHICAGO, ILLINOIS " 
1920 

Reprinted from the Psychological Monographs, Vol. XXVIII, No. 7, 

Whole No. 129. 



ACKNOWLEDGMENTS 

I am much indebted to Professors James R. Angell and 
Harvey A. Carr for numerous helpful suggestions and for gen- 
erous a:id stimulating interest at all stages of the work. My 
thanks are also due to those who kindly gave their time to serve 
as subjects. 



CONTENTS 

I. Introduction i 

II. Technique and Procedure 3 

A. Description and Analysis of Problems Presented to 
Subjects for Solution 3 

B. Apparatus and Recording of Data 10 

C Order of Presentation of Problems 11 

D. Degree of Learning and Uniformity of Procedure 13 

E. Instructions to Subjects 16 

F. The Subjects 17 

III. Results 19 

A. Unit of Measurement 19 

B. Changes in the Rate and Character of Progress. . 22 

C. Perceptual Solutions 35 

D. Analysis 37 

1. Types of Elements Abstracted: Direction of 
Analysis 38 

a. Frequency of Repetition as a Factor 41 

b. Effect of Nearness to a Goal 47 

c. Effect of Speed of Reaction 49 

2. Explicitness and Extent of Analysis 53 

3. Time Relations of Manipulation and Analysis 56 

4. Summary 62 

E. Generalization 63 

1. Relative Absence of Generalization in Percep- 
tual Stage 63 

2. Development of the Concept of the Critical 
Number 64 

3. Random Hypotheses 73 

4. Summary 81 



F. Transfer 82 

1. Degree of Transfer 83 

2. Conditions of Transfer 89 

a. Objectively Identical Elements 89 

b. Subjectively Identical Elements 90 

c. Generalized Methods of Procedure 91 

d. Effect of Thoroughness of Learning upon 
Transfer 93 

3. Negative Transfer 95 

Summary 98 

G. Effect of Age and Education 99 

IV. Discussion and Conclusions 103 

A. Abstraction of Elements 103 

B. Combination of Elements no 

C. Application of Knowledge 116 

D. General Conclusions 120 



I. INTRODUCTION 

The present investigation is an experimental study of the 
mental processes involved in the solution of certain novel prob- 
lems and in the utilization of experiences so gained for the sub- 
sequent mastery of other problems of a similar nature. The 
problems were chosen with a view to the possibility of accurate 
measurement of the progress in their mastery. They are ar- 
ranged roughly in the order of increasing complexity, and are 
so related that the solutions for later problems are, for the most 
part, simply more generalized statements of solutions for earlier 
ones. In the mastery of these problems all forms of mental 
process, from very simple acts of perception to fairly difficult 
and complicated acts of abstraction and generalization, were in- 
volved. It is these latter processes chiefly to which attention is 
directed in the following pages. 

In its use of objective methods the study is closely related to 
the large experimental literature in the field of animal learning 
and the so-called lower forms of human learning. Its domi- 
nant interest in the processes of abstraction and generalization 
brings it into relation with the researches of Kiilpe, Mittenzwei, 
Griinbaum, Moore, Aveling, and Fisher in this field. ^ Its rela- 
tion to the more dynamic studies of Cleveland" and Ruger is 

^ Kiilpe, Oswald, "Versuche uber Abstraktion," Ber. uber den I Kong. f. 
exp. Psych, in Geissen, 1904. 

Mittenzwei, Kuno, "Uber abstrahierende Apperception," Psych. Stud. 
1907, 2. 

Grunbaum, A. A., "Uber die Abstraktion der Gleichheit," Arch. f. d. ges. 
Psychol. 1908, 12. 

Moore, Thomas Verner, "The Process of Abstraction: An Experimental 
Study," University of California Pubs, in Psych., Vol. I. 

Aveling, Francis, "The Consciousness of the Universal." 

Fisher, Sara Carolyn, "The Process of Generalizing Abstraction," Psych. 
Rev., Mon. Sup., Vol. XXI, No. 2, 1916. Fisher summarizes the work in 
generalization and abstraction prior to 1916. 

2 Cleveland, A. A., "The Psychology of Chess and Learning to Play It," 
Am. Jour. Psych., Vol. 18, 1907. 



4 ' JOHN C. PETERSON 

When a single problem of this sort is solved, the solution is 
usually couched in more or less specific terms which do not 
readily function in the subsequent solution of other similar prob- 
lems. In the solving- of a series of related problems, hov^ever, it 
is possible to observe the gradual abstraction of common ele- 
ments and the association of these elements with appropriate 
terms, leading finally to the formulation of a general principle 
for the solution of all problems of the series. If the mastery of 
14 as the initial number of beads constitutes the first problem, 
mastery of 15 as the initial number will constitute the second 
problem, mastery of 16 as the initial number the third, and so on 
until enough problems have been solved to permit the subject 
to develop a general formula for his guidance in drawing from 
any number of beads. ^ 

In the first problem the subject learned that he could not win 
if required to draw from 3, 6, 9, or 12 beads. He also learned 
that when the initial number of beads was 14, he coitld win by 
drawing so as to compel his opponent to draw from these "criti- 
cal" numbers. In the later problems of the series he discovers 
that these numbers are critical (i.e., the one who must draw 
from them inevitably loses the game) in all problems of the 
series, regardless of the initial number of beads presented for 
solution. Here also he generalizes to the effect that all multiples 
of 3 are critical numbers, and that he can win any number which 
is not a multiple of 3 by reducing it to such a multiple at his first 
draw and through successively lower multiples of 3 to o at sub- 
sequent draws. Aside, then, from the development of specific 
responses for the solution of individual problems, the learning 
process here consists in the formation of ( i ) a general concept 
through which the essential elements of all problems may be 
represented by a single term and treated as a unit, and (2) a 

2 For the sake of uniformity it is necessary to fix upon a definite degree 
of mastery of problems to be required in all cases. Some difficulty arises 
here owing to the fact that it is impossible for the subject to win when 
the initial number of beads presented for solution is a multiple of 3. We 
have considered such problems "solved" when the subject expressed a con- 
viction that he could not win and called for a new problem. All other prob- 
lems have been considered solved when two consecutive trials were won. 



HIGHER MENTAL PROCESSES IN LEARNING 5 

system of draws under the control of this concept, by means of 
which all non-critical numbers of the series may invariably be 
won by the subject. The basic concept here developed may be 
termed the critical-number concept. Its functioning and further 
development through use may be observed by requiring the sub- 
ject to solve some additional series of problems somewhat simi- 
lar to those described above. ^ 

New series of problems of this sort may be had by varying the 
numbers of beads which may be taken at a single draw. For 
example, the extension of the numbers which may be drawn to 

1, 2, or 3 (instead of only i or 2 as above) yields a new series 
of problems in which the critical numbers are multiples of 4. 
Further extension of the numbers which may be drawn to i, 2, 
3, or 4 beads yields another series of problems in which the 
critical numbers are multiples of 5, etc. By further changes in 
the numbers which may be drawn an indefinite number of simi- 
lar series of problems may be obtained. The various series of 
problems thus obtained will hereafter be designated by the lowest 
and the highest numbers which may be drawn. Thus Series 
1-2 will denote the series of problems in which only i or 2 beads 
may be taken at a draw; Series 1-3, the series in which i, 2, or 3 
beads may be drawn ; Series 2-3, the series in which only 2 or 3 
beads may be drawn, etc. 

It will be worth while to direct our attention to the location 
of critical numbers in the various series since these numbers 
furnish the key tO' the solution of the various problems and series 
of problems. We have observed already that all multiples of 
3 are critical numbers in Series 1-2. A clearer insight into the 
reason why these numbers are critical in this series will facilitate 
the search for critical numbers in other series of problems. In 
Series 1-2 any pair of draws may result in the removal of either 

2, 3, or 4 beads (the possible combinations of S's and E's draws 
being i-i, 1-2, 2-1, 2-2, and the sum of these combinations 2, 3, 

3 A series of problems was considered solved when the subject gave an 
adequate general formula for the solution of all problems of the series or 
in some other manner clearly indicated his ability to solve all new problems 
of the series at sight. 



6 JOHN C. PETERSON 

3, and 4 respectively). At any draw either S or E can take 
away such a number as to make the sum of his own draw and 
the immediately preceding draw of his opponent equal to 3 by 
taking i when the opponent takes 2, and 2 when the opponent 
takes I. Thus either S or E can cause the number of beads to 
be reduced by 3 with each successive pair of draws, beginning 
with any draw of his opponent, but he cannot force a reduction 
by 2's or 4's in this manner for if the opponent's draw is 2, the 
sum of the pair of draws will necessarily be greater than 2, and 
if the opponent's draw is i, the sum of the pair will necessarily 
be less than 4. The only number, therefore, by which successive 
reductions can thus be forced is 3, and it is clear that 3 is under 
control in this manner only because it is the sum of the low and 
the high draw. If, then, we designate the low draw by the let- 
ter L and the high draw by H, we may say in more general 
terms that all multiples of the sum of L -\- H are critical num- 
bers.^ Notice that these numbers are critical not merely because 
they are multiples of the sum of L + H but because they are 
greater than o by exact multiples of this sum, and o is critical 
by definition. If the conditions of the game were so altered as 
to make i instead of o a critical number, the critical numbers 
would be represented by the terms of an arithmetical progression 
in which the common difference is L + H and the first term is 

* That this formula holds for all series of problems of this sort wherein 
winning consists in securing the last draw, can be shown by substituting the 
general terms, L and H, for i and 2 in the foregoing statement. Thus 
either S or E can take away such a number of beads at any draw as to 
make the sum of his own draw and the opponent's immediately preceding 
draw equal to L + H. If one's opponent draws L, one can draw H; if the 
opponent draws H, one can draw L, and if the opponent draws L -|- x, one 
can bring the sum of the pair of draws up to L -j- H by taking away H — x 
beads. Moreover, L -)- H is the only number by which successive reductions 
can thus regularly be forced for if the opponent's draw is L, one cannot 
draw enough to raise the sum of the pair of draws above L -|- H, and if 
the opponent's draw is H, one cannot draw so low a number as to make 
the sum of the pair of draws less than L -|- H. 

^ This condition is actually realized if in Series 1-2, for example, the defi- 
nition of success is so modified that drawing last constitutes losing instead 
of winning the game. 



HIGHER MENTAL PROCESSES IN LEARNING 



In all series of problems in which L is 2 both o and i may be 
said to be critical by definition. The critical numbers are here 
represented by the terms of two arithmetical progressions hav- 
ing each a common difference of L + H. The first terms of 
these progressions are o and i respectively. When L is 3, 2 
also becomes a critical number. In fact, it will be noticed that 
all numbers below L, including o, are in every series critical. 
These numbers may be called the basic critical numbers. Each 
basic critical number is the first term of an arithmetical pro- 
gression in which the common difference is L + H and of which 
all terms are critical numbers. The critical numbers of any 

series are therefore o, 1,2 L — i and all numbers which 

are greater than any of these basic critical numbers by exactly 
a multiple of the sum of L + H. This generalization holds for 
all series in which all numbers between L and H may be drawn. 
Series of this sort have been called continuous series. All series 
of problems described in the foregoing pages are continuous 
series. Other series of problems requiring more difficult gen- 
eralizations may be had by restricting the draws to L and H. 
Series of this sort in which only L and H may be drawn will be 
known as discontinuous series. Only a limited number of dis- 
continuous series were solved by our subjects, owing to the diffi- 
culty of manipulation when the values of L and H are high. 
To facilitate their analysis these discontinuous series with some 
of their critical numbers are listed below. Bear in mind that 
the numbers by which a discontinuous series is indicated are the 
only numbers which may be drawn in that series. 



Series (discontinuous). 



Critical Numbers. 



I or 3 
I or 5 

1 or 7 

2 or 6 

2 or 10 

3 or 9 

4 or 12 
I or 4 
I or 6 

1 or 8 

2 or 5 



o, 2, 4, 6, 8, 10, 12, 14, 16, etc. 

0, 2, 4, 6, 8, 10, Ji>, 14, 16, etc. 

0, 2, 4, 6, 8, 10, 12, 14, 16, etc. 

0,1, 4,5, 8,9, 12,13, i6,iy, 20,21, etc. 

0,1, 4,5, 8,9, 12,13, 16,17, 20,21, etc. 

0,1,2, 6,7,8, 12,13,14, 18,19,20, 24,25,26, etc. 

0,1,2,3, 8,9,10,11, 16,17,18,19, 24,25,26,27, etc. 

0, 2, 5, 7, 10, 12, 15, 17, 20, 22, etc. 

o, 2, 4, 7, 9, II, 14, 16, 18, 21, etc. 

0, 2, 4, 6, p, II, 13, 15, 18, 20, etc. 

0,1, 4,-, 7,8, II,—, 14,15, 18,—, etc. 



8 JOHN C. PETERSON 

2 or 7 ; 0,1, 4, 5, 9,10, 13,14, 18,19, 22,23, etc. 

2 or 8; 0,1, 4,5, /o,/i, 14,15, 20,21, 24, 25, etc. 

2 or 9; 0,/, 4,5, 8,-, 1/,/^", IS, 16, 19, — , 22,23, etc. 

3 or 7; 0,/,^, 6,-,-, 10,11,12, 16, — , — , 20,21,22, etc. 

3 or 8; 0,7,.?, 6,7,-, 11,12,13, 17,18, — , 22,23,24, etc. 

4 or 9; o,7,^,j, 8,-,-,-, 13,14,15,16, 21,—, — ,— , etc. 

The numbers in italics are those obtained by applying our old 
formula to these series. These numbers will be known as pri- 
mary critical numbers and groups of them as primary C groups. 
Within the intervals between primary C groups there are now 
other critical numbers which have become critical because of 
the restriction of draws to L and H. Such numbers will be 
known as secondary critical numbers. The secondary critical 
numbers fall into order in accordance with a principle already 
applied to the primary critical numbers; namely, that there are 
certain basic critical numbers each of which is the first term of 
an arithmetical progression in which the common difference is 
L + H and of which all terms are critical numbers. Our task 
will therefore be confined to finding the basic secondary critical 
numbers in the series listed above; i.e., those secondary critical 
numbers which fall in the interval between the first two groups 
of primary critical numbers. 

Note that in every series in the list all numbers below H, which 
may be obtained by adding a multiple of 2L to any nWmber in 
the primary C group, are critical. Why are these numbers criti- 
cal ? and why do they not extend beyond H — i above o ? Ob- 
viously the possibility of using the high draw no longer exists 
when the number of beads has been reduced below H. Below 
this point the game must necessarily proceed by successive re- 
ductions of L beads at each draw, and each pair of draws will 
result in the reduction of the number by 2L. Now if, after 
reduction below H, the number of beads remaining is equal to 
any number in the primary C group plus a multiple of 2L, it 
will itself be critical, because the one who draws first in the first 
pair of draws will necessarily draw first in all succeeding pairs 
of draws, wherefore the last and winning draw will fall to his 
opponent. Concerning the second question, it is clear that any 
number which is greater than o, or than any other member of 



HIGHER MENTAL PROCESSES IN LEARNING 9 

the basic primary C group, by exactly the amount of H, cannot 
be critical; because it can he reduced by one draw to less than 
L, i.e., to a basic critical number. This is the reason why 4, 6, 
and 8 are not critical numbers in Series i or 4, i or 6, and i or 
8 respectively. This is the reason also why the secondary C 
groups sometimes consist of less than L critical numbers, as in 
Series 2 or 5, 2 or 9, 3 or 7, 3 or 8, and 4 or 9. Each of the 
numbers in these series, which would have been critical but for 
the limiting effect of the H draw, is indicated in the list by a 
dash. All numbers that are greater than a basic secondary criti- 
cal number by an exact multiple of L -|- H, are themselves 
critical for the same reason that numbers which are greater than 
a basic primary critical number by that amount, are critical. 

To recapitulate, we have found certain uniformities in the 
distribution of critical numbers in our series of problems as fol- 
lows : (i) There is in ez^ery series a group of critical numbers 
extending from o to L minus i inclusive, which we have called 
the basic primary C group. (2) All numbers in any series, which 
are greater than any number in the basic primary C group by 
exactly a multiple of the sum of L + H, are critical. (3) In 
the discontinuous series all numbers below H which are greater 
than any number in the basic primary C group by exactly a multi- 
ple of 2L, are critical. These are the basic secondary critical 
numbers. (4) All numbers in a series, which are greater than 
a basic secondary critical number by exactly a multiple of the 
sum of L -]- H, are critical.*^ The primary critical numbers are, 
found in all series, both continuous and discontinuous ; secondary 
critical numbers are found only in discontinuous series. To win 
a game, or trial, beginning with any number which is not criti- 
cal, one must draw so as to reduce the number of beads to a 
critical number at his first draw ; thereafter he must draw so 
as to make the sum of each of his draws and the preceding 
draw of the opponent, equal to L -f- H.'' 

6 Other more difficult series of problems may be obtained by permitting 
three or more non-consecutive numbers or groups of numbers to be drawn. 
But the foregoing series furnish enough difficulty for the ordinary gradu- 
ate student under the conditions of our experiment. 

7 In the discontinuous series this procedure may result in the final re- 



10 JOHN C. PETERSON 

B. Apparatus and Recording of Data 
The apparatus consisted of a stop-watch, a metronome, an 
Edison telescribe, or dictaphone, and a row of thirty beads 
strung upon a steel wire the ends of which were inserted into 
metal cubes of sufficient height to permit the beads to move 
freely. These beads were substituted for the matches of the 
original game because they offered less obstruction to free ma- 
nipulation, and therefore increased the value of the time records 
of individual reactions. 

While the experiment progressed the dictaphone was in ac- 
tion with the horn so adjusted as to secure as clear a record as 
possible of all the verbal reactions of the subject. The subject 
was asked to state aloud the number taken at each draw as the 
"move" was being made. The experimenter also called out the 
number taken by him at each draw. Thus the dictaphone records 
contained all the draws and such comments as were made during 
the experiment. To mark the time consumed by these reactions 
a metronome beating half seconds was placed near the dicta- 
phone. The beats were clearly audible in the records. As the 
experiment progressed the experimenter made a written record 
of the individual draws of the subject and of the experimenter 
in separate columns. Incipient movements, indicating that the 
subject was considering a particular draw, were recorded with a 
distinctive mark but were not counted in the summing up of 
data. Upon these written records the dictaphone records were 
later transcribed. The beats of the metronome were counted, 
and the number of half seconds from the beginning of the trial 
to each of its component reactions were recorded. Though no 
high degree of accuracy is claimed for these time records, they 
were accurate enough to be exceedingly useful in showing the 
distribution of attention among the different reactions entering 
into a trial. They also serve to show differences in the speed of 
reaction of different subjects and of any one subject in different 
portions of the experiment. From the dictaphone records were 

duction of the number of beads to a basic secondary critical number, 
which will be reduced to a basic primary critical number by a series of 
L draws. 



HIGHER MENTAL PROCESSES IN LEARNING ii 

also obtained the comments of subjects, which were recorded 
in their proper places in the records. These verbal reactions 
were often useful in the interpretation of our data. In order 
that the learning process might go on as naturally and as free 
from interruption as possible, no introspections were called for 
and few were given. 

For various reasons it was found necessary in certain por- 
tions of the work to dispense with the use of the dictaphone 
except for the recording of the more important verbal responses. 
Time measurements were here obtained by means of a stop- 
watch. In such cases it was of course impossible to record the 
time of each separate reaction, although the time of complete 
trials was taken with as great accuracy as was possible by the 
more cumbersome method. With this procedure it was not 
possible to distinguish the time consumed by the experimenter's 
draws from that consumed by the reactions of the subject. To 
make the records of different subjects comparable in time, there- 
fore, the experimenter made the time of his draws as uniform 
as possible. A fair estimate of the degree of their uniformity 
may be made from the record of a relatively small number of 
typical reactions. In 206 trials with 11 beads taken from the 
records of various subjects, it was found that the average time 
per trial was 41.14 metronome beats, or half seconds. The aver- 
age time per trial consumed by the experimenter's draws was 
9.76 half seconds with an average deviation of 1.99. This A.D. 
of 20 per cent from the average of the experimenter's time per 
trial is not a serious matter when it is remembered that his 
time was on the average less than one-fourth of that consumed 
by the reactions of the subject. The effect of the inclusion of 
the time required for the experimenter's draws is to put the 
subject who draws most rapidly at a slight disadvantage, but 
the time differences in our records are large enough to make 
these small errors rather unimportant. 

C. Order of Presentation of Problems 
As already stated, each series consists of what are to the 
subject at the beginning a succession of relatively independent 



12 JOHN C. PETERSON 

problems. Beginning with H + ^ beads in all series in which 
L is I, and with H + 2L — i beads in all series in which L is 
greater than i, the numbers (or problems) of each series were 
presented in the order of their magnitude from lowest to high- 
est. As soon as a practicable solution was found for one num- 
ber, the next higher number of the series was presented. Thus 
successive problems of a series were solved until a satisfactory 
generalization was formulated for the entire series. A higher 
series of problems was then presented in the same manner. The 
order of presentation of series was the order of the magnitude 
of their L and H draws. First, all series in which L is i were 
presented, beginning with Series 1-2 and proceeding through 
Series 1-3, 1-4, 1-5, etc., tmtil a general solution for all series 
of this order^ was found. Series of the next higher order (i.e., 
in which L is 2) were next presented, beginning with Series 2-3 
and continuing through Series 2-4, 2-5, etc., until a general so- 
lution for all series in which L is 2, was found. In the same 
manner successively higher orders of series were presented until 
a general solution was obtained for all continuous series. The 
number of continuous series was not the same for all subjects. 
A wor'kable generalization was made by some subjects after the 
solution of but a few series in only two or three orders. Others 
were unable to arrive at a suitable generalization until after the 
solution of numerous series extending through a number of 
orders. Only a limited number of discontinuous series was pre- 
sented, however, and this number was the same for all subjects. 
These discontinuous series are listed on page 7 in the order in 
which they were presented. 

The order of presentation of the discontinuous series may 
need a word of explanation. The first portion of this group of 
series, extending from Series i or 3 to Series 4 or 12 inclusive, 
was given to bring out the fact that secondary critical numbers 
are found by the addition of multiples of 2L to the primary 
critical numbers. This is the simplest possible collection of 
series with a sufficient variety of values for L and H, in which 

^ The order of a series refers to the number represented by L. In series 
of the first order L is i, in the second order L is 2, etc. 



HIGHER MENTAL PROCESSES IN LEARNING 13 

all multiples of 2L above any primary critical number, are criti- 
cal. Not all subjects succeeded in getting this generalization 
from this small group of series, but the addition of other series 
of this character was impracticable owing to the large number 
of beads that would need to be manipulated. In Series i or 4 
to 4 or 9 it was expected that subjects would discover that not 
all multiples of 2L above a primary critical number, are critical ; 
and also find some rule by which to locate such multiples of 2L 
above primary critical numbers as are not, in a given series, 
critical. 

After the completion of Series 4 or 9 the beads were put 
aside, and the subject was asked to name the critical numbers in 
Series 4 or 1 1 and later in Series 5 or 11. The work in these 
series was necessarily done mentally and often served to compel 
the subject to return to his earlier generalizations in the at- 
tempt to find a principle which would apply here. If at the 
end of these series no single workable generalization for all 
series had been given, the subject was permitted to refer to a 
table of critical numbers for all the discontinuous series.^ Here, 
as in the preceding portions of the work the time required and 
such of the subject's observations as were given verbal expres- 
sion, were recorded. 

D. Degree of Learning and Uniformity of Procedure 

In order to make the conditions uniform for all subjects it 
was necessary that the experimenter govern his draws in ac- 
cordance with some definite rules. It was, of course, necessary 
that no rule should prevent the experimenter from taking ad- 
vantage immediately of any error made by the subject, but aside 
from this there was a fair latitude within which he might vary 
his draws without necessarily affecting the outcome of the game. 
In Series 1-2, for example, so long as the subject took care to 
reduce the number of beads to a multiple of 3 at each of his 
draws, it might seem to be a matter of indifference whether the 
experimenter took i or 2 at a draw. But the indiscriminate 
changing of his draws would result in a serious lack of uni- 

^ This table with a few changes in the italics is given on page 7. 



14 JOHN C. PETERSON 

formity in the condition to be met by the different subjects. If, 
on the other hand, the experimenter were to take i always or 2 
always so long as the subject's draws were correct the latter 
might, by mere mechanical memory and repetition of previous 
draws, win a sufficient number of consecutive trials to permit 
him to pass on to the next higher number in the series, and some 
mechanical performance of this sort might be repeated indefi- 
nitely. To prevent such mechanical repetition of accidental suc- 
cesses while still maintaining uniform conditions for all subjects, 
the experimenter first drew i at every draw through an entire 
trial and then 2 at every draw throughout the next trial. This 
change in the number drawn by the experimenter in alternate 
trials was continued until the problem was solved. But if S made 
an error in any trial, E immediately departed from his uniform 
reactions and, taking advantage of the error, drew so as to win. 
The change in E's draw in alternate trials often proved very dis- 
concerting for a time to subjects who tried to repeat accidental 
successes from memory, but it usually resulted in bringing about 
some kind of attempt at critical analysis. When two trials had 
been won in succession by the subject, another bead was added 
and the process repeated. Thus successive increments of i 
were made and the solutions for the resulting numbers found 
until a solution for the entire series was obtained. 

This changing back and forth of the experimenter's draw in 
alternate trials, together with the requirement that S win two 
trials in succession, resulted in bringing out all possible varia- 
tions in Series 1-2. Two lines of procedure each somewhat in 
accord with this were possible in the higher series. (i) E 
might alternate the L draw in one trial with the H draw in the 
next, neglecting all the intermediate draws except when an error 
on the part of the subject made it possible for E to win with 
one of the intermediate draws. (2) He might use all of the 
possible draws, taking each exclusively for an entire trial in 
some regular order. The latter procedure was used with some 
of the subjects in the preliminary experiments. It has the advan- 
tage of preventing, in a measure, the direction of attention to 
non-essentials. But, if consistently carried out, it requires that the 



HIGHER MENTAL PROCESSES IN LEARNING 15 

subject win each number in the series as many times as there are 
possible draws in the series and so interferes seriously with the 
comparability of data from different series. The requirement 
that S win H minus L consecutive trials with each number also 
becomes somewhat monotonous in the higher series. The former 
procedure has the advantage of avoiding this monotony and 
keeping the degree of learning more nearly uniform through- 
out all series. It therefore makes the data from different series 
more truly comparable. This procedure was followed in the 
wofk of all of our subjects except a few who served in the 
preliminary experiments. No attempt is made to compare their 
work in the later series with that of other subjects. ^° 

As soon as a subject recognized two successive critical num- 
bers in a series by inspection, or showed his mastery by predic- 
tion of the outcome with higher numbers in the series, or gave 
a correct generalization, he was presented with a considerably 
higher number in the same series. He did not draw from this 
higher number, but merely announced his decision as to what 
would be the correct draw and gave a reason for his decision. 
In case the correct draw was indicated but no intelligible reason 
could be given, the subject was taken back to the point previously 
attained and required to follow the regular procedure until clear 
evidence was given of at least some sort of a practical solution 
which would hold for all numbers in the series. This was not 
a very common occurrence, however, and never resulted in 
serious departures from the ordinary procedure. In most cases 

i<> Even this procedure cannot, however, be said to make the data from 
different series entirely comparable. The fact that the critical and the non- 
critical numbers are not in the same proportion in. different series, is bound 
to affect the degree of learning in some measure. Compare Series 1-2 with 
Series 1-5 in this respect. In the former series the subject encounters five 
critical numbers in working from the beginning of the series to and in- 
cluding 15 beads; in the latter he encounters two. In the former series he 
need not reduce the number of beads to each of these critical numbers more 
than four times before passing to the next higher critical number. In the 
latter series he must reduce the beads to each of the critical numbers at 
least ten times before he is permitted to pass on to the next higher critical 
number. There is, of course, in the more frequent occurrence of critical 
numbers in the lower series, some compensation for the higher degree of 
learning of critical numbers in the higher series. 



i6 JOHN C. PETERSON 

a verbal formula was given, and sometimes this was expressed in 
mathematical form. 

At the completion of a series the next higher series followed 
immediately or at the next sitting. The solving of successive 
series continued until a general solution for an entire order of 
series was given or until a complete series was solved without 
a draw. When this occurred, a higher "test" series of the same 
order was usually but not always given, being omitted only when 
the subject's statement of his generalization left no doubt that 
he had found an adequate solution for all series of that order. 
In case of failure in one of these test series, it need hardly be 
said, a return was made to the series following the last one which 
had been solved. If the test series was successfullly solved, the 
first series of a higher order was presented, and so on through- 
out the experiment. As already stated, it was impossible to 
adhere strictly to this uniform degree of learning in case of the 
discontinuous series. Here the degree of learning differed con- 
siderably with different subjects, some of whom persistently 
worked out the relations existing between successive series 
whereas others were content to generalize for each series in 
isolation. 

E. Instructions to Subjects 
With the apparatus in position and the subject ready to begin 
work, the experimenter said in substance: "We shall draw 
alternately from this string of beads. You may take i or 2 
beads at a draw and I also may take i or 2 at any draw. (With 
each statement of the number to be drawn the experimenter il- 
lustrated by manipulation of the beads, showing the alternation 
between his and the subject's draws which were made from op- 
posite ends of the row of beads.) The object is to get the last 
draw, i.e., you win if you get the last draw. You always draw 
first. Will you call out the number you take at each draw?" 

The number was then immediately reduced to 4 and the sub- 
ject asked to begin. The procedure from this point has already 
been described. As soon as the subject announced his inability 
to win with 6 beads as the initial number, the experimenter 



HIGHER MENTAL PROCESSES IN LEARNING 17 

asked : "Hereafter whenever you make such a discovery or get 
an idea which seems to be signifcant, will you let me know at 
once?" The subject was also told at this point, provided his 
questions had not brought out the fact earlier, that he would be 
expected to find a general rule for the solution of all numbers 
in the series. The request that S make known any seemingly 
relevant new ideas was repeated at the beginning of each of the 
early series and at other times when his actions seemed to in- 
dicate that he had made a new discovery. Subjects were not 
permitted to record any data though the experimenter's record 
for a few of the immediately preceding trials was usually in 
sight. This was unavoidable owing to the difficulties of manipu- 
lation and recording. While attempting to recall successful re- 
actions of earlier portions of the work, subjects sometimes asked 
permission to see their records. This was not permitted. At 
the beginning of the second series the subject was told : "The 
requirements of the game are the same as before but you may 
now take either i, 2, or 3 beads at a draw." The subjects were 
of course informed at the beginning of each of the later series 
as to the numbers which might be drawn. Subjects were asked 
to refrain from thinking of the experiments during the intervals 
between their periods of work. 

F. The Subjects 

Exclusive of those who served in the preliminary experimen- 
tation by which the final mode of procedure was determined, 46 
subjects served in our experiments. The subjects were divided 
into groups as follows: (a) Group I consisting of 14 subjects, 
(b) Group II consisting of 20 subjects, (c) Group III consisting 
of 12 subjects. The major portion of our study is based upon 
the work of the subjects in Group I. These subjects will be 
designated by Roman numerals. Subjects i to x are numbered 
in the order of the speed of their performance as determined by 
the number of trials required for the solution of all the problems 
presented, Subject i being the speediest. Subject ii the next speed- 
iest, etc. Subjects xi, xii, xiii, and xiv did not finish the entire 
experiment and could not, therefore, be given their relative 



i8 JOHN C. PETERSON 

ranks. Subject iii was a first-year high school boy who was thir- 
teen years old; Subject vi was a college senior, and the remain- 
ing ones were all graduate students or instructors in the Uni- 
versity of Chicago. All of the subjects of this group were 
males except Subject v. 

Subject ii solved most of the problems at two sittings in one 
day and returned three weeks later to finish in one short period. 
Subjects i, iii, iv, vi, vii, and xii were scheduled to work every 
day, but failed to live completely up to the schedule. Subjects 
ix and x worked with a fair degree of regularity at weekly 
intervals, and the remaining subjects of this group were quite 
irregular in the distribution of their effort. The work with this 
group of subjects was done at the University of Chicago during 
the fall and winter of 1916-17. 

Group II consists of 20 members of a class of 21 third- and 
fourth-year college and graduate students at the Kansas State 
Agricultural College. This group is subdivided into Groups 
Ila and lib each consisting of 10 subjects. Group Ila solved 
first Series 1-2 and then Series 1-3. The members of this group 
are designated by the capital letters A to J in the order of their 
speed of learning, the speediest first. Group lib solved first 
Series 1-3 and then Series 1-2. The members of this group are 
designated by the small letters from a to j, again in the order of 
their speed of learning. The work with Group II was done at 
the Kansas State Agricultural College in the spring of 19 18. 

Group III consists of all of the members of a class of third- 
and fourth-year college students. The subjects in this group 
worked out solutions for problems in Series 1-2 from 14 to 20 
inclusive. The work with this group of subjects was done at the 
Kansas State Agricultural College during the fall of 1918. 



III. RESULTS 
A. Unit of Measurement 

Before proceeding with the presentation of data it is neces- 
sary to attend to the selection of a suitable unit for the measure- 
ment of progress. To be entirely comparable successive units 
of effort should be uniform in (a) time consumed, (b) degree 
and distribution of attention, and (c) number and character of 
reactions. Perfect uniformity in all of these characteristics is 
obviously out of the question, if subjects are to be given the 
requisite freedom for normal reasoning. There is no constant 
unit of time into which the work of our subjects can be divided 
so that the distribution of effort in successive units will conform 
even approximately to (b) and (c) of the foregoing require- 
ments. Of a number of units of response which suggest them- 
selves the trial^ conforms most closely to all of these require- 
ments.^ Measurement and comparison of progress will there- 
fore be made chiefly in terms of the trial, though the number of 
errors made during the solution of the various problems, as well 
as the time required, will also be stated and utilized to some ex- 
tent in the treatment of some portions of our data. 

It will be worth while briefly to inquire into the variability of 
the trial. The characteristics in which variability is greatest and 
also most susceptible to accurate measurement, are the time and 

^A series of draws culminating in the reduction of any initial number of 
beads to o, constitutes a trial, regardless of who draws last. 

2 It might be supposed that the draw would be a better unit of measure- 
ment, but this unit is altogether too variable to be of service. In time con- 
sumed it varies from less than a half second to several hundred seconds. 
In the amount of critical attention called forth, it varies from serious, con- 
centrated effort to the most mechanical performance. Often, indeed, the 
draws do not stand out singly as significant to the subject, but are linked up 
in various combinations into series each of which constitutes a trial and a 
real unit in the distribution of attention. Nor can the measurement of 
progress be made in terms of errors or of successful performances, — i.e., 
of successes in reducing the number of beads to o, — since in drawing from, 
a critical number no successes of this objective sort and likewise no erro- 
neous draws, are possible. 



20 



JOHN C. PETERSON 



number of draws per trial. The facts regarding the variability 
of trials in these characteristics, as found in the work of our 
major group of subjects in Series 1-2, are given in Table I. 



3V 









Table I 










No. of 


Subject's 
Av. No. 


Draws per 


Trial 


T 


ime 


per 


Trial 


5Ubj. 


Draws 


A.D. 


V3 


Med. Time 


Q^ 


V5 


iii 


3-13 


•95 


.30 


27 




7.0 


.26 


iv 


3-33 


.71 


.21 


39 




131 


•34 


i 


3-35 


.67 


.20 


41 




11.7 


.29 


V 


3.58 


.91 


•25 


58 




30.0 


.52 


ii 


3-66 


.89 


.24 


85 




34-9 


.41 


viii 


4.20 


.68 


.16 


25 




5-4 


.22 


vii 


4.21 


1.41 


.34 


27 




10.5 


•39 


xi 


4-25 


1.02 


.24 


54 




16.2 


•30 


ix 


431 


1.29 


.29 


26 




6.8 


.26 


xiii 


S-oi 


1.68 


•33 


25 




9-5 


•38 


xii 


5.22 


1.62 


•31 


48 




26.5 


•55 


X 


6.84 


2.89 


.42 


43 




14.6 


•33 


vi 


7.15 


1.68 


.23 


36 




10.3 


.29 


xiv 


8.03 


364 


•45 


33 




14-5 


•44 


A.D. 




40 —Z— 


-Q^ 


5\7 — . 


Q 







Av. 



Med. 



The table is self-explanatory. Note the slight tendency for 
subjects who rank high in the number of draws per trial to rank 
low in time per trial. The average of the average numbers of 
draws per trial for the first seven subjects listed is 3.64 and 
the average of their median times per trial is 21.5 seconds. The 
corresponding averages for the last seven subjects listed are 5.83 
draws and 16.9 seconds. The correlation between these two lists 
of averages is — •105,® The variations in time and in number 
of draws per trial do not, therefore, tend to show any cumulative 
effect upon the relative values of trials in the work of different 
subjects. Moreover, the distribution of attention is such as 
largely to discount the value of variations in time and number 
of draws per trial — especially the latter. It seldom occurred that 
a subject attended carefully to all of the draws in a trial when 
the initial number of beads was higher than 17 or 18. One 
portion of the trial was usually regarded as crucial and the at- 

6 Rank method. 



HIGHER MENTAL PROCESSES IN LEARNING 21 

tention mainly directed upon that point with the result that all 
but a few of the draws in the trial were executed with only a 
minimum of attention. If these mechanical draws were here 
neglected, the variation in number of draws per trial would be 
extremely slight, and the time variations would actually tend to 
compensate for the variations in the number of draws per trial/ 
Again, after the longer pauses, subjects frequently came back 
tO' the work declaring that they had permitted their minds to 
wander in the pursuit of irrelevant associations. The comments 
of subjects indicate that such distractions occurred rather fre- 
quently. If these distractions are to be regarded as indications 
of a low degree of attention, we have here another case of an 
inverse or compensating relationship between variations in dif- 
ferent aspects of the trial. Such practice value as the mechani- 
cal draws may have possessed was perhaps completely offset by 
their distracting effect upon attention to points which were re- 
garded as crucial. At any rate this was the spontaneous verdict 
of a number of the subjects. 

The variability of trials within Series 1-2 is typical of that in 
all later series in which any considerable difficulty was encount- 
ered. There was no great variation in the average length and 
duration of trials from series to series. In the later series — 
especially those which were presented late in any group of series 
of the same order^ — there were generally fewer draws per trial, 
hnd attention tended to be more evenly distributed among the 
various draws of the trial, though mechanical draws practically 
always appeared in any series where the number of draws per 
trial ran high. The average time per trial required by Subjects 
i to X inclusive in some of the more difficult series, is given in 
Table II. The series reported in this table represent approxi- 
mately two-thirds of all trials of these subjects in the entire 
experiment. 

'' This compensating relation in the variations of time and number of 
draws per trial is not, of course, found in the trials upon higher as com- 
pared with those upon lower numbers of a series in the work of the same 
subject. Here the time per trial usually varies directly vi^th the number of 
draws though not in the same proportion. But even here it is doubtful 
whether the longer trials are of more value than the shorter ones, owing 
to the presence and distracting effect of the more mechanical draws. 



22 JOHN C. PETERSON 

Table II 



Series 


No 


. Trials 


Av. Time per Trial^ 


A.D. 


V. 








(in seconds) 








1-2 




2o6o 


24.09 




546 


.23 


2-3 




, 647 


42.72 




10.68 


.25 


2 or 6 




360 


43-15 




23.69 


•55 


I or 4 




423 


38.42 




14. 1 1 


•37 


2 or 7 




265 


42.19 




2.94 


.07 


8 These averages 


and deviations are those 


of all trials in a 


series without 


regard to the 


average time 


per trial of individual 


subjects. 





There is a general tendency for subjects to exercise a little 
more caution after the solution of one or two series than at first. 
This accounts for the marked increase in the average time per 
trial in Series 2-3 as compared with that of Series 1-2. In the 
last four series listed the uniformity of the average time per 
trial is fair, but the deviations are large. However, it should 
not be overlooked that in the more important matter of degree 
and distribution of attention the variability, though not sus- 
ceptible to quantitative statement, is undoubtedly much less 
marked, and that the variations in one aspect of a trial often 
tend to compensate for these in other aspects. On the whole, 
it is safe to say, the trial in the present experiment is not less 
uniform than the units of measurement regularly employed in 
studies on maze learning. Its uniformity is probably far greater 
than that of the trials in certain experiments in ball-tossing, 
from which curves of learning have been plotted and conclusions 
of far-reaching consequence drawn. 

B. Changes in the Rate and Character of Progress 

The progress of the ten subjects of our major group, who 
solved all of the series of problems, can be traced in Table III. 
The series of problems are listed at the left in the first column. 
Following each series are the records of the work of individual 
subjects upon it. In columns P and E are given the number of 
problems (i.e. of different numbers within a series) which are 
solved and the number of errors made in arriving at a general 
solution of the series. The time is given in seconds. 



HIGHER MENTAL PROCESSES IN LEARNING 



23 













Table III 












Progress of 


Subjects through all Series of Problems 










•Subj 


ect i 






Subject ii^ 




Subject iii 




Series 


P 


E Trials Time 


P 


E 


Trials Time 


P 


E Trials 


Time 


1-2 


10 


24 


48 


1716 


12 


8 


35 1572 


12 


5 


28 


464 


1-3 


4 





7 


247 


9 


4 


23 585 


5 


I 


10 


120 


1-4 


5 


5 


14 


124 























i-S 











19 























2-3 


22 


II 


44 


1661 


8 


3 


17 507 


II 


I 


25 


427 


2-4 











10 


4 





4 215 


28 


13 


84 


168s 


2-5 




















38 


16 


13 


60 


1115 


2-6 
































32 


3-4 











165 








53 











IS 


3-5 











123 























4-5 











215 








SO 











45 


I or 3 


4 


— 


4 


667 


8 


— 


8 269 


7 


— 


II 


"5 


I or 5 


12 


— 


13 


1387 


2 


— 


3 118 


6 


— 


13 


173 


I or 7 





— 





54 





— 


36 


4 


— 


5 


62 


2 or 6 


17 


— 


21 


2483 


3 


— 


3 178 


15 


— 


30 


618 


2 or 10 





— 





483 


4 


— 


4 20I 


12 


— 


18 


289 


3 or 9 


7 


— 


6 


381 


5 


— 


6 437 





— 





65 


4 or 12 





— 





120 





— 


193 





— 





14 


I or 4 


4 


I 


8 


492 


12 


15 


38 2701 


20 


6 


47 


757 


I or 6 











46 


24 


I 


47 4076 


14 


I 


29 


467 


I or 8 











23 


II 


I 


19 636 











193 


2 or 5 











34 


II 





14 606 


6 





9 


361 


2 or 7 











32 


12 





15 644 


16 


I 


19 


835 


2 or 8 











342 

















164 


2 or 9 











78 


II 





15 647 











55 


3 or 7 











35 

















80 


3 or 8 











21 


II 





15 664 











27 


4 or 9 











38 








310 











90 


4 or II 








40 






36s 








25 


5 or II 








30 






230 








62 


General solution 




















for all 


series 






155 






533 








70 


Total 


85 


41 


165 


11215 


147 


32 


266 15864 


172 


41 


388 


8425 



9 Series 2 or 8 and 3 or 7 were accidentally left out of the experiment with 
this subject. 



24 



JOHN C. PETERSON 











Table III 


(Continue 


d) 














Subject i\ 






Subject 


V 




Subject vi 




Series 


P 


E Trials Time 


P 


E 


Trials Time 


P 


E Trials 


Time 


1-2 


II 


55 


96 


2205 


13 


23 


62 


2834 


27 


221 


331 


7934 


1-3 


12 


13 


35 


704 


8 


I 


15 


363 


II 


6 


30 


265 


1-4 


10 


6 


27 


449 


2 





3 


33 


2 





4 


54 


1-5 


6 





10 


87 











9 











40 


1-6 


14 





24 


241 


























1-7 


8 





14 


157 


























1-8 


o 








36 


























2-3 


13 


16 


44 


1800 


21 


19 


71 


4566 


15 


5 


30 


709 


2-4 


6 


I 


13 


279 


13 


7 


29 


894 


I 





I 


268 


2-5 


o 








30 











54 














3-4 


o 








25 











17 


I 








205 


3-5 


o 




















28 











74 


4-5 


o 








26 











10 











75 


I or 3 


7 


— 


17 


180 


4 


— 


4 


52 


10 


— 


10 


237 


I or 5 


5 


— 


9 


no 


10 


— 


18 


377 


14 


— 


18 


503 


I or 7 


2 


— 


3 


33 


10 


— 


14 


269 


13 


— 


14 


235 


2 or 6 


27 


— 


45 


1818 


33 


— 


46 


1189 


24 


— 


36 


821 


2 or ID 


6 


— 


II 


546 


25 


— 


32 


916 


II 


— 


14 


262 


3 or 9 


4 


— 


6 


151 





— 





72 


13 


— 


13 


258 


4 or 12 


o 


— 





75 





— 





165 


9 


— 


8 


142 


I or 4 


12 


7 


26 


430 


13 


3 


21 


392 


19 


I 


31 


484 


I or 6 


13 


5 


24 


1006 


23 


5 


47 


1657 


6 





8 


147 


I or 8 


o 








58 


16 


5 


28 


984 


2 





2 


66 


2 or 5 


o 








152 


8 





8 


427 


16 


2 


21 


346 


2 or 7 


8 





14 


590 


21 


2 


34 


163 1 


6 





6 


145 


2 or 8 


2 





2 


333 


7 





10 


527 


3 





4 


75 


2 or 9 


O 








58 


10 





15 


993 











66 


3 or 7 


17 


6 


33 


1526 











285 


8 





10 


159 


3 or 8 











600 











192 


3 





2 


58 


4 or 9 


o 








604 











40 











I6S 


4 or II 








820 








85 








65 


5 or II 








55 








290 








35 


General 


solution 






















for all ! 


series 






8610 








1210* 








280 


Total 


183 


109 


453 


23794 


227 


65 


457 


20561 


214 


235 


594 


14173 



HIGHER MENTAL PROCESSES IN LEARNING 



25 











Table III 


(Continue 


d) 














Subject 


vii 




Subject viii 




Subj 


ect i>. 




Series 


P 


E Trial 


5 Time 


P 


E 


Trials Time 


P 


E Trials 


Time 


1-2 


19 


182 


371 


6659 


15 


90 


168 


2591 


28 


187 


315 


6129 


1-3 


12 


3 


23 


754 


12 


24 


56 


684 


6 


4 


16 


244 


1-4 


5 





8 


207 


10 


8 


30 


294 


2 





3 


23 


1-5 











15 


I 





2 


23 


I 





2 


52 


1-6 























15 


4 





9 


115 


1-7 



































6 


2-3 


18 


I 


25 


1827 


69 


27 


167 


5408 


42 


25 


119 


5859 


2-4 


9 





13 


1047 


35 


9 


71 


2679 


25 


4 


53 


1956 


2-5 











28 


15 





20 


500 


10 


I 


15 


295 


2-6 














2 





2 


59 











35 


2-7 























9 














3-4 


8 





12 


301 


19 


4 


41 


699 


9 





10 


155 


3-5 











(^2, 


3 





4 


75 


6 





9 


118 


3-6 


19 


I 


30 


3115 











13 











20 


3-7 











21 


























4-5 











12 











21 














I or 3 


6 


— 


12 


372 


24 


— 


22 


1067 


17 


— 


36 


IIOI 


I or 5 


8 


— 


II 


327 


19 


— 


24 


870 


5 


— 


II 


195 


I or 7 





— 





6 


3 


— 


4 


76 


I 


— 


2 


52 


2 or 6 


19 


— 


19 


2427 


23 


— 


38 


1268 


51 


— 


108 


4189 


2 or 10 


7 


— 


16 


979 


29 


— 


34 


755 


17 


— 


20 


489 


3 or 9 





— 





60 


19 


— 


20 


309 


14 


— 


18 


680 


4 or 12 





— 





80 





— 





52 





— 





87 


I or 4 


14 


I 


22 


1583 


24 


2 


39 


877 


37 


41 


154 


7230 


I or 6 


12 


I 


19 


4691 


12 





16 


303 


14 





24 


1318 


I or 8 











30 











40 


14 





21 


947 


2 or 5 


II 


2 


20 


728 


24 


8 


46 


1441 


II 


I 


19 


758 


2 or 7 


7 





12 


614 


48 


12 


86 


3702 


12 


3 


27 


966 


2 or 8 


7 





13 


954 


9 


2 


14 


263 


9 


2 


13 


446 


2 or 9 


8 





10 


1 102 


12 


I 


IS 


271 


10 





15 


660 


3 or 7 


7- 


2 


13 


293 


26 


2 


34 


677 


10 





21 


360 


3 or 8 


8 





8 


678 


II 





14 


369 


ID 





17 


369 


4 or 9 


4 





5 


828 


8 





9 


278 


II 


I 


22 


589 


4 or II 








455 








690 








240 


5 or II 








1 105 








910 








ISO 


General 


solution 






















for all 


series 






4980 








2420 








7740* 


Total 


208 


193 


662 


36341 


472 


189 


987 


29708 


376 


269 


1970 


43753 



26 



JOHN C. PETERSON 



Table III (Continued) 
Subject X 



Series 


P 


E 


Trials 


Time 


Series 


P 


E 


Trials 


Time 


1-2 




47 


313 


597 


17538 


2 or 


10 


7 


— 


7 


233 


1-3 




16 


3 


45 


1 122 


3 or 


9 


27 


— 


28 


980 


1-4 




2 








136 


4 or 


12 


9 


— 


8 


514 


1-5 
















I or 


4 


19 


II 


37 


1306 


2-3 




36 


16 


105 


4881 


I or 


6 


22 


14 


49 


1421 


2-4 




15 





26 


728 


I or 


8 


26 


5 


z(> 


1973 


2-5 




3 





2 


77 


2 or 


5 


14 


3 


21 


1016 


2-6 
















2 or 


7 


35 


4 


50 


1922 


3-4 




37 





40 


240 


2 or 


8 


9 





12 


377 


3-5 




I 








230 


2 or 


9 


I 








118 


z-(> 




10 





12 


AAA 


3 or 


7 


19 


I 


21 


968 


317 













28 


3 or 


8 


12 


I 


14 


548 


4-5 




II 





10 


570 


4 or 


9 


12 


2 


18 


485 


5-6 













50 


4 or 


nil 






327 


I or 


3 


21 


10 


23 


858 


5 or 


II 








193 


I or 


5 


8 


— 


10 


466 


General 


solution 






I or 


7 


3 


— 


4 


225 


for 


all 


series 






1960= 


2 or 


6 


13 


— 


14 


545 


Totals 


435 


373 


1 192 


42470 



The most striking feature of these figures is the great ir- 
regularity in the number of trials, the time required, etc., for 
the solution of various problems. Note the relatively large num- 
ber of trials required in Series 1-2, and the rapid decrease in the 
number of trials in successive series of the first order.^^ The 
same sort of change is found in series of the second order and, 
to some extent, in all higher orders of the continuous series as 
well as in those of the discontinuous series. These regular and 
more or less gradual changes are due in part to the separate 
mastery of numerous relatively isolated though often poorly 
defined "elements" of the problematic situation; in part they are 
due to a gradual definition and development of larger and more 
complex units in the form of concepts and general principles. ^^ 

10 No errors are possible in those series where blanks appear in the 

error column. 

"Subjects were required to work Series 4 or 11 and 5 or 11 "mentally." 
* These subjects failed to find a general solution for all series in the time 

at their disposal. 

12 The order to which a series belongs is determined by the value of L. 
In series of the first order L is i ; in those of the second order L is 2, etc. 

13 Any aspect or isolable portion of the objective situation and any rela- 
tionship between such portions or aspects is here regarded as an element 
of the problem. 



HIGHER MENTAL PROCESSES IN LEARNING 27 

These gradual changes are sometimes obscured by irregular and 
often extremely abrupt changes arising from various causes, 
such as, distraction of attention, sudden utilization of old con- 
cepts, monotony, fatigue, rest, elimination of erroneous assump- 
tions, change of methods, etc. 

The more striking irregularities in the rate of progress are 
shown in graphic form in Figure I. The number of trials and 
the amount of time required as well as the number of errors 
made by each of the ten subjects in various series and groups of 
series, is represented by the columns in the figure. The series 
and groups of series represented in the figure are indicated at 
the left. The subjects are designated by the numerals at the base 
of columns. 

This irregularity is in some degree illusory owing to the 
failure of some bonds in the early stages of their development 
to register their effect in terms of those particular overt responses 
which were taken into account in the score. That bonds do 
begin to develop long before their effect is apparent in the crude 
score, is evident from the comment of subjects, the comparative 
length of delays, incipient reactions, etc., some of which are in- 
cluded in the detailed records and will be noted later. Further 
evidence of the inadequacy of any one sort of measure to regis- 
ter all of the changes in the strength of bonds, is found in the 
relation between the time, the errors, and the number of trials 
involved in the solution of successive series of problems. In 
Figure I the number of errors made and the amount of time 
consumed in the solution of the problems in each section of the 
figure are shown by the relative height of the first and third 
columns respectively. Observe that in every case in Series 1-2 
the error column is higher than the time column, and that there 
is a somewhat gradual shifting in the relative height of these 
columns so that in the last group of series the time column is 
in every case higher than the error column.^* This change is to 

1* The relation between the trial column and the time and error columns in 
Series 1-2 and in the last group of series, is worthy of note. In every case 
but one the height of the trial column is intermediate between that of the 
other two columns. The measurement of progress in terms of trials there- 
fore takes special account of the conflicting claims of time and errors where 
the conflict is greatest. 



28 



JOHN C. PETERSON 




Figure I. — The Roman numerals at the base of the columns of the lowest 
section represent the various subjects of Group I, who solved all series. The 
height of the middle column for each subject represents the number of 
trials required by him for the solution of the problems indicated by the num- 
ber at the left. The trial columns are all directly comparable. The num- 
ber of errors made and the amount of time required by each subject are 
represented by his first and third columns respectively. The time and error 
columns are drawn to such a scale for each subject as to make their total 
length for all series equal to the total length of the trial columns of that 



HIGHER MENTAL PROCESSES IN LEARNING 29 

some extent influenced by increased caution, as is shown by the 
lengthened average time per trial in the later series as compared 
with the first series; but it is due mainly to the fact that per- 
formance represents a distinctly lower plane of learning here 
than does formulation. It is a little surprising to find this rela- 
tion between power of performance and power of formulation 
here in view of the simplicity and definiteness of the terms which 
are required for formulation. It is possible that if success were 
to demand a high speed of reaction, as is generally the case in 
acts of skill and to a large extent in practical thinking, this re- 
lationship would tend to disappear. However, with the time 
allowed for each reaction limited only by his patience, the sub- 
ject is usually able to respond correctly while he is yet unable 
to foretell the correct response in the absence of the appropriate 
concrete situation or to say why the response is correct when 
made. This disparity in power of performance and power of 
formulation as well as some of the more striking cases of change 
in speed of progress, both of the regular and of the irregular 
sort, will receive more special attention in later sections. 

The principal features of the progress from series to series 
are duplicated in the progress from problem to problem in some 
of the more novel series. There are, however, some important 
differances. Series 1-2 being the first and most novel series il- 
lustrates the natural progress from problem to problem better 
than do any of the later ones. The progress of the fourteen 
subjects of the major group through the problems of this series, 



particular subject. They are thus directly comparable with one another and 
with the trial columns of the same subject. But the error and time columns 
of different subjects are not directly comparable. 

The numbers at the left represent various series and groups of series of 
problems as follows : 

1. Series 1-2. 

2. All later continuous series in which L is i. 

3. Series 2-3. 

4. All later continuous series in which L is 2. 

5. All continuous series in which L is 3 or greater. 

6. All discontinuous series from Series i or 3 to Series 4 or 12 inclusive 

7. Series i or 4. 

8. All remaining discontinuous series. 



30 JOHN C. PETERSON, 

is shown in Table IV. The problems of the series are listed at 
the left in the column headed "Initial No. of Beads." The num- 
ber of trials and of seconds required by each subject for the so- 
lution of each problem are listed in self-explanatory form. 

The errors are not included in this table but their number is 
roughly proportional to the number of trials per problem except 
in the critical numbers, where no errors are possible. Certain 
points of difficulty for various subjects are revealed by the large 
number of trials and the amount of time required for the solution 
of some of the problems. Such points of unusual difficulty are 
found in the record of Subject iv at lo beads, in that of Subject 
nI at II beads, and in that of Subject vi at 25 beads, etc. Some- 
times the troublesome problems are so distributed as to suggest 
delnite, regularly recurring types of difficulty. This is illustrated 
in the record of Subject vi where numbers which are equal to i 
plus any multiple of 3 offer greater difficulty than those which 
are equal to 2 plus a multiple of 3. Problems of the former 
type will be referred to as L-problems and those of the latter type 
as H-problems. It will be noticed that the L-problems from 7 
to 22 inclusive required 72 trials for their solution whereas the 
H-problems from 8 to 23 inclusive required only 25 trials. The 
solution of 25 beads which is an L-problem, required but 6 trials. 
Only in two instances did this subject solve an L-problem with 

Table IV. — Progress of Subjects through Series 1-2 

Initial No. Subj.i Subj. ii Subj. iii Subj. iv Subj. v 

of Beads Trials Time Trials Time Trials Time Trials Time Trials Time 



4 


2 


21 


2 


29 


3 


42 


4 


32 


2 


20 


5 


2 


13 


2 


29 


2 


24 


4 


28 


2 


17 


6 


4 


48 


2 


35 


3 


36 


3 


44 


2 


32 


7 


2 


25 


2 


97 


3 


27 


7 


117 


6 


89 


8 


4 


59 


3 


96 


4 


56 


4 


79 


4 


78 


9 


5 


86 


3 


70 


I 


21 


9 


148 


II 


240 


10 


5 


104 


2 


79 


3 


50 


50 


1151 


7 


256 


II 


18 


719 


6 


375 


3 


81 


4 


152 


II 


577 


12 


4 


102 


3 


112 


2 


42 





15 


2 


117 


13 


4 


539 


2 


143 


2 


32 


8 


270 


3 


106 


14 


. 





7 


355 


2 


29 


3 


169 


7 


647 


15 








I 


152 





24 








2 


252 


16 


























3 


406 


otal 


48 


1716 


35 


1572 


28 


464 


96 


2205 


62 


2834 



HIGHER MENTAL PROCESSES IN LEARNING 31 

Table IV (Continued) 

Initial No. Subj. x Subj. ix Subj. vi Subj. vii Subj. viii 

of Beads Trials Time Trials Time Trials Time Trials Time Trials Time 



4 


4 


35 


4 


22 


2 


4 


4 


40 


2 


15 


5 


4 


42 


2 


14 


2 


4 


4 


68 


2 


21 


6 


4 


94 


9 


85 


8 


33 


22 


190 


7 


60 


7 


4 


62 


3 


25 


II 


87 


74 


533 


2 


16 


8 


4 


77 


3 


25 


5 


46 


4 


35 


4 


36 


9 


17 


.292 


4 


35 


7 


56 


12 


343 


8 


88 


10 


8 


108 


4 


44 


5 


45 


48 


664 


15 


158 


II 


24 


339 


68 


822 


6 


68 


II 


145 


32 


395 


12 


28 


423 


15 


192 


8 


86 


26 


335 


18 


193 


13 


100 


2067 


6 


94 


IS 


211 


12 


156 


33 


559 


14 


9 


204 


76 


1455 


2 


30 


18 


257 


24 


551 


15 


44 


1229 


3 


54 


7 


lOI 


36 


590 


12 


264 


16 


48 


1 165 


88 


1799 


ID 


151 


II 


237 


3 


104 


17 


48 


1220 


3 


126 


7 


108 


21 


428 


3 


91 


18 


15 


369 


3 


76 ^ 


5 


84 


16 


567 


3 


40 


19 


21 


687 


3 


135 


26 


468 


7 


202 








20 


21 


446 


2 


72 


3 


42 


24 


538 






21 


3 


58 


I 


A7 


10 


154 


2 


265 


167 


2591 


22 


38 


1054 


4 


248 


5 


76 


9 


1064 






22 


3 


67 


2 


82 


2 


28 










24 


6 


154 


2 


34 


7 


123 


371 


6659 






25 


5 


121 


2 


137 


156 


5061 










26 


3 


78 


2 


84 


6 


126 










27 


8 


264 


I 


48 


10 


450 


Sub 


. X (Continued) 


28 


2 


68 


2 


114 


4 


100 


No. of Beads 


Trials 


Time 


29 


9 


262 


2 


152 


2 


22 




41 


2 


100 


30 


5 


141 





19 





' 157 




42 


2 


114 


31 


10 


337 


I 


87 










43 


7 


480 


32 


6 


244 














44 


3 


237 


33 


ID 


325 






331 


7934 




45 





HID 


34 


5 


181 


315 


6129 








46 


18 


I2I2 


35 


2 


95 












47 


2 


151 


36 


5 


187 












48 





19 


37 


17 


696 












49 


6 


467 


38 


2 




77 

Tex 












SO 


4 


400 


39 


3 


151 


















40 


7 


284 














597 


17538 



Table IV (Continued) 



il No. 


Subj. 


xiv 


Subj. 


xii 


Subj. 


xiii 




Subj. xi 


Beads 


Trials 


Time 


Trials 


Time 


Trials 


Time 


Trials Time 


4 


4 


18 


2 


17 


2 


16 




2 9 


5 


4 


42 


2 


9 


3 


13 




3 40 


6 


3 


59 


5 


58 


5 


49 




I 31 


7 


3 


44 


6 


69 


4 


26 




2 25 


8 


5 


•86 


7 


137 


3 


22 




2 51 


9 


II 


124 


8 


99 


6 


49 




I 33 


lO 


24 


352 


17 


295 


8 


63 




7 182 


II 


36 


568 


4 


116 


30 


338 




8 225 


12 


12 


123 


9 


156 


38 


481 




2 53 


13 


4 


52 


II 


250 


2 


31 




13 419 


14 


2 


14 


28 


1016 


4 


89 




2 53 


IS 


10 


131 


7 


256 


6 


103 




I 8 


i6 


5 


93 


39 


1622 


10 


183 




2 56 


17 


14 


230 


5 


393 


3 


38 




2 57 


i8 


10 


150 





127 


5 


125 




41 


19 


4 


16 


5 


253 


2 


38 




3 83 


20 


9 


254 


4 


211 


7 


217 




2 55 


21 


12 


2X8 





63 


2 


96 




60 


22 


5 


78 


4 


170 


5 


132 







23 


7 


152 


2 


97 


2 


105 




— 


24 


3 


51 





2 


2 


108 




53 1481 


25 


13 


295 


3 


114 


4 


132 






26 


2 


38 


2 


65 


2 


70 






27 


4 


68 





I 


2 


216 






28 


7 


146 


2 


152 


3 


87 






29 


2 


37 


2 


58 


2 


76 






30 


3 


62 





88 


I 


27 






31 


14 


355 


2 


91 


3 


136 






32 


6 


214 


2 


74 












33 


2 


68 





22 










34 


25 


758 


2 


49 


166 


3066 






35 


2 


74 


2 


39 










36 


2 


72 





5 


Subj. 


xiv (Continued) 


zy 


23 


793 


2 


106 


No. Beads Trials 


Time 


38 


5 


117 








50 




3 


81 


39 





I 






4 




2 


II 


40 


7 


zn 


184 


6283 


5 




2 


9 


41 


3 


134 






6 







2 


42 


I 


97 






7 




5 


84 


43 


18 


^1^ 






8 




4 


47 


44 


2 


60 






9 




I 


16 


45 





4 






10 




3 


23 


46 


4 


149 






II 




3 


58 


47 


4 


148 






12 







3 


48 





6 






13 




2 


14 


49 


4 


116 






14 
15 
16 

17 




2 

2 
2 


17 

I 

9 
24 



367 8I7I 



HIGHER MENTAL PROCESSES IN LEARNING 33 

fewer trials than were required for the preceding H-problem, 
A total of 236 trials was required for the solution of 9 L-prob- 
lems and only 35 for 9 H-problems. Subject x required 296 
trials for the solution of the L-problems of this series and only 
146 for an equal number of H-problems. Subject xiv experi- 
enced a change in the type of difficulty at about the middle of 
the series. The L-problems from 7 to 22 inclusive recjuired only 
45 trials whereas the H-problems from 8 to 23 inclusive required 
73 trials. But from this point to 44 beads the L-problems be- 
came the more difficult, requiring 107 trials as compared with the 
22 trials required for the solution of an equal number of H-prob- 
lems in this portion of the series. A similar change in the type 
of difficulty occurs in the record of subject vii. The L-problems 
below 12 required 126 trials while the H-problems were solved 
in 19 trials. Yet the numbers of trials required for the solution 
of an equal number of L- and H-problems above this point were 
30 and 83 respectively. 

The principal causes of these irregularities are found to be 
false assumptions; erroneous generalizations, founded usually 
upon very incomplete analysis; uncritical applications of methods 
and generalizations from preceding problems, and, in some of the 
extreme cases, monotony and fatigue. Some of these irregulari- 
ties will be considered at greater length in later sections on gen- 
eralization and transfer. 

Another noteworthy feature of the progress of most of the 
subjects through the problems of Series 1-2 is the low number of 
trials upon the early and the late problems of the series as com- 
pared with the number of trials required for the solution of some 
of the intermediate problems. The extent to which this feature 
characterizes the work of various subjects is shown in Figure H. 
The number of trials is represented on the abscissae and the num- 
ber of problems solved — or recognized as insoluble in case of 
the critical numbers — is represented on the ordinates. Note that 
with the exception of the curve of Subject iii, all curves show 
a period of relatively rapid advancement followed by one of 
slower progress, after which a final period of rapid progress 
terminates with a successful generalization for the entire series. 



34 



JOHN C. PETERSON 



O31';OdM0idaQ:]3i]Kl 







HIGHER MENTAL PROCESSES IN LEARNING 35 

The three stages in the learning- process represented by these 
features of the curves are also clearly indicated by the behavior 
of the subjects and by their comments. The period of rapid ini- 
tial rise of the curves was obviously a period of perceptual solu- 
tions. During the intermediate period of slower advancement 
the abstraction of significant elements of the problematic situa- 
tion occupied the greater portion of the energies of the subjects. 
Some combination among these elements of course occurred im- 
mediately, but it was not until after considerable experience with 
the separate elements that most subjects were able to combine 
and organize them into effective means of control, capable of 
altering markedly the direction of the curve and leading speedily 
to a satisfactory generalization for the entire series. The types 
of mental process which in some degrees dominated these various 
stages of learning may profitably be treated separately in the 
three following sections. 

C. Perceptual Solutions 
At the beginning of Series 1-2 comparatively few erroneous 
draws were made, but the percentage of errors increased rapidly 
for some time with the increase in the number of beads. This 
increase in the percentage of errors is shown in Table V. This 
table includes all of the reactions of thirteen subjects upon the 
non-critical numbers from 4 to 11 inclusive. 

Table V 

Number of Beads 4 

Total Draws from each Number 222 

Erroneous Draws from each Number 14 
Percentage of Errors 6.4 

Note the rapid increase in the percentage of errors as the num- 
ber of beads advances from 4 to 8 and the high but fluctuating 
percentage beyond 8. The unevenness of this increase is also 
worthy of note. From 5 beads the percentage of erroneous draws 
is 17.3 per cent higher than from 4; from 8 beads it is 17.6 per 
cent higher than from 7. But the percentage of erroneous draws 
from 7 is only 4. i per cent higher than from 5 though the inter- 
val between 5 and 7 is twice as wide as that between the members 



5 


7 


8 


ID 


II 


194 


282 


262 


297 


401 


46 


79 


119 


92 


173 


23-7 


27.8 


454 


30.9 


431 



36 JOHN C. PETERSON 

of the other pairs of numbers compared. This difference is un- 
doubtedly due to the nature of the errors which are possible in 
drawing from the different numbers. The only errors possible 
in drawing from 4 and from 5 beads result in leaving 2 and 4 
beads respectively; from 7 and 8 the only possible errors result 
in leaving 5 and 7 beads respectively. Thus from the point of 
view of the number of beads left after an erroneous draw from 
each of the foregoing numbers, the long intervals occur where 
the change in the percentage of errors is greatest. The change 
in the percentage of errors in the early numbers of the series is 
therefore apparently due to the degree of ease with which the 
consequences of a given move may be foreseen in a purely per- 
ceptual manner.^^ 

The perceptual character of these early solutions is further 
shown by the behavior of the subjects. Usually i or 2 beads 
were moved aside tentatively or otherwise marked off so that the 
possible result of further draws could be directly perceived. ^^ 
This tentative manipulation was purely a trial and error affair 
at first, as was indicated by the constant shifting from one of 
the possible draws to another, as well as by the comments of 
those subjects who mentioned the matter at all. Sometimes both 
hands were employed in this sort of manipulation,' — one repre- 
senting the subject and the other the experimenter, — and so the 
consequences of all possible draws were figured out on the per- 
ceptual level. The movements used to mark off, or temporarily 
exclude some of the beads from attention, were not always of this 
overt character. Often they became almost imperceptible and 
occasionally verbal reactions served this function even in the 
early stages of the game. For some subjects these perceptual 
judgments were important factors in the solution of problems 
far beyond the immediate span of attention. Thus some of the 

15 The large downward fluctuation in the percentage of errors at 10 offers 
no serious difficulty to this view. This fluctuation occurs where it might 
most reasonably be expected; i.e., immediately after one of the long inter- 
vals and after the range of direct perceptual control has been passed. 

16 If the hold upon the beads thus tentatively drawn aside was released 
the maneuver was counted a draw; therefore the caution, and also the diffi- 
culty of applying perceptual checks upon the higher numbers. 



HIGHER MENTAL PROCESSES IN LEARNING 37 

subjects, upon discovering late in the game that 6 or 9 could not 
be won, would mark off and exclude so many beads from at- 
tention and then proceed to solve the remaining numbers upon 
the perceptual level. But usually when the number of beads was 
increased to beyond 7 or 8, the perceptual form of solution more 
or less completely broke down. 

The important feature of these perceptual solutions is the fact 
that all progress was here made through trial and error per- 
ceptually checked and entirely without the use of symbols except 
an occasional word used in a very specific way. That is to say, 
there was a conspicuous absence of generalization in these early 
perceptual solutions, due to the lack of any necessity for the use 
of symbols. As will be seen later, the failure to utilize symbols 
in these early problems seriously limited the transference of con- 
trol from the lower to the higher problems of the series. 

D. Analysis 

Immediately following upon the period of perceptual solutions 
there was usually a period of evident confusion. Though sub- 
jects had been instructed at the beginning of the series to look 
for underlying principles, they generally failed up to this point 
of the game to see any relation between problems and to remem- 
ber how or understand why certain numbers had been won. In 
accordance with James's view that all analysis depends upon the 
"law of varying concomitants" or upon the elements having 
somehow previously been brought to attention in isolation, fur- 
ther progress here would require either that the solutions of 
earlier numbers be recalled and applied to the problems at hand 
or that some sort of manipulation be carried on by means of 
which the significant elements of the various problems could be 
abstracted and associated with appropriate reactions. Both of 
these alternatives were tried, the former without success in a 
single case when attempted early in Series 1-2, the latter with 
varying degrees of success, depending on the individual who 
made the attempt and the sort of manipulation resorted to. 

The sorts of analysis which occurred are classified from the 
points of view of (i) the specific elements and types of elements 



38 JOHN C. PETERSON 

which were abstracted and employed for generalization, (2) the 
explicitness and extent of analysis, and (3) the temporal rela- 
tions of manipulation and ideational analysis/^ 

I. Types of Elements Abstracted: The Direction of Analysis. 

Any aspect or isolable portion of the objective situation and 
any relation between such portions or aspects will hereafter be 
referred to as an "element" of the general problem. Elements 
of this sort appeared in considerable variety and exhibited uni- 
formities of such varied types that different subjects might con- 
ceivably have arrived at equally valid solutions from quite differ- 
ent lines of approach. 

Subjects usually began early in Series 1-2 to count the number 
of beads from which they were required to draw at each move. 
This usually led to an early discovery of the fact that certain 
numbers are especially significant as points of orientation and 
control in the series. Certain multiples of 3 were usually the first 
numbers to take on this special significance in Series 1-2. Thus 
subjects almost invariably came to regard the numbers which 
they could not win as the important elements to be sought out. 
Sometimes subjects also tried to remember the numbers which 
they were able to win, but these numbers were seldom made the 
objects of special attention and the basis of hypotheses and 
generalizations, except as they were brought in to complete the 
formulation after the solution had been practically worked out 
upon some other basis. 

Perhaps the next most common type of element to attract 
special attention was the relationship of the draws made by the 
experimenter to those of the subject. Occasionally a subject 
would become so absorbed in the pursuit of this relation that he 
would utterly fail for a time to notice the significance of the 
number of beads presented. Most of our subjects, however, 
noticed some sort of relationship here rather early and divided 
their attention between it and the number element mentioned 
above. 

Other elements of particular interest at times to various sub- 

1^ See Ruger, "The Psychology of Efificiency," pp. 10-14. 



HIGHER MENTAL PROCESSES IN LEARNING 39 

jects were the number of beads obtained by the subject or by the 
experimenter or both, or sometimes the total number of beads 
drawn, or the number of draws obtained by the experimenter or 
by the subject, or the relation between the number of draws and 
the number of beads obtained, etc. Though elements of this 
sort all exhibit uniformities of such character as might well be- 
come the basis for successful generalizations, no one succeeded 
in getting a successful solution from these elements alone, al- 
though some subjects lost a considerable amount of time in the 
attempt. 

Without going further into detail it may be said that the more 
successful subjects usually began very early to give special at- 
tention to the number of beads remaining after each draw and 
often failed entirely to notice the relation between the draws of 
the subject and those of the experimenter. Often in the first 
series, and quite generally in the later series, elements of both 
of these types were constantly taken into account with good re- 
sults. The attention of the less successful subjects usually fluctu- 
ated considerably between the different types of elements, but 
failed to follow up any type consistently enough to discover the 
uniformities lying beneath the surface. 

Table VI shows the point in the first series where each subject 
gave the first evidence of having become definitely aware of the 
uniformities in the elements of each of the first two types men- 
tioned above. This, of course, implies a considerable amount of 
previous attention to the elements underlying the uniformities. 
For example a subject would often study his own successful 
draws and those of the experimenter for some time before be- 
coming aware of the "opposite"^^ relation between them; or he 
might realize for a considerable time that certain definite num- 
bers are critical without noticing that they are multiples of 3. 

It will be noticed that a number of the most successful sub- 
jects seemingly failed to discover the principle of drawing by 

^^ In all trials beginning with a critical number E drew i when S drew 2, 
and 2 when S drew i. Likewise, in order to win any non-critical number, 
S had to draw the "opposite" of E's preceding draw throughout the trial 
after first having reduced the number of beads to a multiple of 3 at his 
first draw. 



40 



JOHN C. PETERSON 









Table 


VI 










Point in 


Series 


1-2 where 


Point in 


Series 1-2 whe 




the principle of 


drawing 


it was 


discovered that 


ibject 


by "opposites" was 




the C 


numbers are 




discovered 




multiples of 3 








20 




I3bi& 50th trial 


•; 










15b 


35" 


( 


11 








iii 


13b 


25 th 


trial 




15b 


28" 


' 


iv 


pb 


25" 


" 




15b 
i6b 
28b 


96" 

63rd 

384th 


i 


V 

vi 


14b 


71st 


trial 




' 


vii 


6b 


21 " 


" 




i8b 


310" 


' 


viii 


i8b 


168" 


a 




17b 


162nd ' 


' 


ix 


23b 


303rd 


" 




31b 


aisth 


i 


X 


12b 


77th 


" 




40b 
2ib 


547" 
53rd 


I 


XI 








xii 


13b 


70th 


trial 




lob 


30th 


' 


xiii 


12b 


86" 


" 




15b 


III " 


( 


xiv 


lib 


88" 


" 




40b 


293rd 


« 



1^ 13b, 15b, etc., stand for the various problems in the series and indicate 
the initial number of beads presented at each trial in the problem. 

20 The principle of drawing opposites was not discovered in this series by 
Subjects i, ii, v, and xi. 

Opposites, and that some of the subjects who found the greatest 
difficulty in solving the problems discovered this principle very 
early in the series. By the rank method the correlation between 
the number of trials required for the discovery of this principle 
and the number of trials required to find a satisfactory solution 
for the series, is — .324. There is a positive correlation of .728 
between the number of trials required to discover that the critical 
numbers are multiples of 3 and the number of trials required to 
find a satisfactory solution for the series. Attention on the part 
of the subject to the numbers of beads from which he must draw 
thus leads more directly to fruitful generalizations than does at- 
tention to the relation of his own draws to those of the experi- 
menter. 

What factors, it may be asked, determine which of the various 
elements shall be abstracted from the total situation in any case? 
This question cannot be answered fully from the data at hand 
but it seems worth while to point out that the order of abstrac- 



HIGHER MENTAL PROCESSES IN LEARNING 41 

tion of elements is exactly what might be expected if the prin- 
cipal selective factors were the relative frequency of occurrence 
of various situation and response elements, the relative nearness 
of the various elements to a goal, or end of action, and the speed 
of the subject's reactions. These three factors will be discussed 
separately in a later section. Some data concerning them may 
well be given here. 

a. Frequency of Repetition. — As already noted the numbers 
in Series 1-2 which were first to attract special attention and to 
become objects of active research were almost invariably the 
critical numbers. This emphasis upon the critical numbers was 
the result not of sudden, comprehensive insight, but of a gradual 
and often tedious growth of meaning. There were, to be sure, 
sudden spurts and long plateaus in the progress of some sub- 
jects, as a result of the drawing in of old concepts by associa- 
tion. But the regularity of progress in the abstraction of ele- 
ments in the first series was not often seriously obscured by these 
factors. The first number to be isolated and treated as especially 
significant was almost invariably 6."^ Thereafter 9, 12, and the 
higher critical numbers followed in the order of their magnitude 
except as the order was affected by the influx of old concepts in 
the form of generalizations. The early stages of the abstraction 
of the various critical numbers progressed with surprising inde- 
pendence. The first objective signs of the process of abstraction 
appeared usually in the form of short delays, exclamations, and 
other indications of critical reaction, which, if we may trust the 
comments of subjects, were accompanied by occasional fleeting 
insights into the nature of the situation. Continued repetition 
of these numbers served to give further emphasis to them and 
greater depth and stability to the erstwhile fleeting insights into 
their signifiance for the solution of other numbers, until finally it 
became possible to formulate their relations into satisfactory 
principles of control. 

21 It is true that prior to the recognition of the critical significance of 6, 3 
was usually recognized in the concrete as a losing number. But, owing 
apparently to the easy perceptual control of 3, it was almost never mentioned 
explicitly as a losing number, except as an afterthought in rounding out a 
generalization which had been made upon the basis of higher critical numbers. 



42 JOHN C. PETERSON 

Corresponding to the gradual abstraction of certain critical 
numbers and the order in which these numbers were affected by 
the process, is the gradually increasing excess of the subjects' 
reactions to critical numbers over their reactions to non-critical 
ones. Not only does this excess of draws from critical numbers 
increase from problem to problem, but the ratio of draws from 
critical to those from non-critical numbers also increases rather 
steadily. The regularity of this increase, both in the excess of 
draws from critical numbers and in their ratio to the draws from 
non-critical numbers, may be seen in Table VII. At the left are 
listed the various problems of the series from the first problem, 
in which 4 beads are presented, to the twelfth problem, in which 
15 beads are presented for solution. The figures in the succeed- 
ing columns indicate the total number of times ten subjects were 
required to draw from each of the numbers listed above the 
columns, in the solution of the problem opposite which the figures 
occur and all preceding problems. Thus in the solution of the 
problem, 4b, the subjects drew 12, 14, o, and 28 times from i, 2, 
3, and 4 beads respectively. In the solution of 4b and 5b they 
drew 23, 26, 5, 28, and 26 times from i, 2, 3, 4, and 5 beads 
respectively, and so on. Or reading down the column, under 3 
for example, the ten subjects drew o times from 3 while work- 
ing upon 4b; 5 times while working upon 4b and 5b; 46 times 
while working upon 4b, 5b, and 6b, and so on. The ratios of 
draws from critical numbers to the averages of those from ad- 
joining non-critical numbers, are given in italics. 

The frequency of subjects' draws from any critical number 
does not greatly exceed that of their draws from adjacent non- 
critical numbers until after the game has progressed to a point 
considerably beyond the critical number in question. Thus the 
frequency of draws from 3 beads becomes greater than from 
2 or 4 beads only after considerable work upon 6 beads; and the 
frequency of draws from 6 beads becomes greater than from 5 
or 7 beads only after some work upon 9 beads, etc. This is in 
entire accord with the fact that the status of these numbers (i.e., 
whether critical or non-critical), was in most cases forgotten 
shortly after the commencement of work upon higher numbers, 



HIGHER MENTAL PROCESSES IN LEARNING 



43 



Table VIII 

Total Number of Draws of ten Subjects from various Numbers in Solving the first 

twelve Problems of Series 1-2. 





I 


2 


3 


4 


5 


6 


7 


8 


9 


10 


II 


12 


13 


14 


15 


4b 


12 


14 





28 






















5b 


23 


26 


5 
.19 


28 


26 










* 












6b 


23 


26 


46 
1.70 


28 


26 


41 

1.58 




















7b 


35 


41 


61 

1-47 


42 


43 


41 
■9^ 


44 


















8b 


44 


54 


77 
145 


52 


58 


55 
1.08 


44 


38 
















9b 


44 


54 


152 
2.8T 


52 


58 


130 

^•55 


44 


38 


75 
1.97 














lob 


65 


69 


241 


78 


79 


195 


89 


88 


75 
.69 


131 












lib 


108 


78 


408 
4.08 


122 


lOI 


333 
2.66 


149 


151 


172 
1.22 


131 


220 










12b 


108 


78 


533 


122 


lOI 


458 


149 


151 


297 
2.11 


131 


220 


125 
.56 








13b 


137 


98 


669 

5.J5 


152 


124 


578 
3-66 


192 


186 


385 
2.03 


194 


291 


125 

■52 


186 






14b 


163 


109 


758 
5-28 


178 


137 


657 


226 


208 


447 
2.01 


236 


339 


161 
.61 


186 


126 




15b 


163 


109 


845 
5.59 


178 


137 


744 
4.09 


226 


208 


534 


236 


339 


248 
.96 


186 


126 


87 
.69 



(See preceding paragraph for explanation of table.) 

and was re-discovered only after a considerable number of trials 
during which the frequency of repetition of forgotten critical 
numbers became gradually more preponderant over that of for- 
gotten non-critical numbers. Thus, after working for a while 
upon 7 or 8 beads, the subject usually forgot that he had been 
unable to win 6 and did not often re-discover the fact until after 
some work upon 9 beads or even upon higher numbers; that is, 
until the number of draws frorn 6 had come to be greatly in 
excess of those from 5 or 7 beads. The behavior of subjects 
toward 9 and 12 beads, and often toward higher numbers, was 



44 JOHN C. PETERSON 

similar in this respect. But, owing to the increasing influence of 
generaHzation and other less conscious sorts of transfer, the pre- 
ponderance of draws from the critical numbers became gradually 
less marked or entirely vanished. 

Though in the work of these subjects the order in which the 
various critical numbers acquired their special significance is 
closely paralleled by the relative frequency of their presentation, 
it might be questioned whether this order was not determined by 
the order of their original presentation as problems of the series 
rather than by the frequency of subjects' reactions to them. 
That the order of the original presentation of these numbers as 
separate problems is not the determining factor, however, is evi- 
dent from the facts presented in Table VIII where the critical 
numbers were not presented as separate problems prior to their 
discovery. The discovery of a critical number was usually not a 
sudden event, but a gradual process of isolation and of growth 
of meaning. The process will be referred to hereafter as the 
abstraction of critical numbers. Various stages of the process 
must be distinguished and defined before quantitative comparison 
becomes possible. The first stage in the abstraction of a critical 
number may be regarded as completed when, with this number of 
beads concretely before him, the subject gives the first clear in- 
dication of his realization that in the current trial defeat is in- 
evitable. This recognition of defeat may be expressed in a shak- 
ing of the head or in some form of exclamation, such as, 'T lose," 
"No use," "You win," "I cannot win now," or "No matter how 
I draw now you can win." Such recognition of defeat often 
comes in the form of a perceptual judgment, without explicit 
awareness of the exact number of beads remaining or of the im- 
possibility of ever winning from this particular number. The 
second stage in the abstraction of a critical number will be re- 
garded as completed when the subject clearly states the recog- 
nition of his inability ez'er to win the number in question. The 
third stage is regarded as completed when the subject announces 
his conviction that the number in question is also critical for the 
experimenter; i.e., that the experimenter must inevitably lose if 
required to draw from that number. In the interpretation of 



HIGHER MENTAL PROCESSES IN LEARNING 45 

reactions indicative of the various stages the general context 
must be taken into account; our criteria are not, therefore, wholly 
objective. 

The individual records of twelve new subjects, reacting to 14 
beads as their initial problem, are given in Table VIII. The sub- 
jects are listed at the left as Si, S2, S3, etc., according to the 
number of trials required to win two trials in succession from 
14 beads. In the / columns of the various A sections (i.e., 3 A, 
6A, 9A, and 12 A) of the table are given the numbers qf draws 
made by individual subjects from each of the critical numbers 
during the first stage of its abstraction. Similar data for the 
second and third stages of abstraction are given in the B and C 
sectons of the table respectively. The totals of draws from the 
two non-critical numbers adjacent to each critical number, dur- 
ing the various stages, are given in the s columns of the respec- 
tive sections. Thus the record of S3 shows that only the first 
critical number was abstracted. During the first stage, 3A, of 
this process two draws were made from 3 beads and none from 

2 or 4 beads. Prior to the completion of Stage 3B, 39 draws 
were made from 3 beads and a total of only 6 from 2 and 4 
beads. Before 3C was completed 74 draws had been made from 

3 beads though the total number of draws from 2 and 4 beads 
was only 16. 

In only 4 of the 73 pairs of /- and .y-column figures here pre- 
sented does the number of draws from a critical number fall 
short of the sum of all draws from the two adjacent non-critical 
numbers, and in no case does it fall below the average number of 
draws from the two adjacent non-critical numbers. The average 
number of draws from critical numbers at the completion of 
Stage A in their abstraction, is 31.2; that of draws from adja- 
cent non-critical numbers is 4.2. At the completion of Stage B 
the corresponding averages are 51.6 and 9.9 respectively; and 
at the completion of Stage C they are 100.6 and 21.8 respectively. 
The total number of draws from all critical numbers in all stages 
of their abstraction as here reported, is 3787. From non-critical 
numbers — though there are twice as many of them as of critical 
numbers — the total number of draws is only 703. That is to 



46 



JOHN C. PETERSON 



Table VIII 

Frequency of Subjects' Draws from the Critical Numbers at the Completion 

of the various Stages of their Abstraction. 





3A 


3 


B 


3C 


6A 


6B 


6C 




/ 


.$■ 


/ 


^ 


/ 


^ 


/ 


J 


/ 


^ 


/ 


J 


Si 


I 


o 


55 


2 


167 


31 


52 


4 


57 


4 


158 


40 


S2 


4 


I 










7 


I 










S3 


2 





39 


6 


74 


16 














S4 


I 





28 


4 






3 





14 


2 






ss 


I 


o 










4 





20 





43 


14 


S6 


I 

















II 


I 






S7 


3 


I 






II 


5 


2 


I 


12 


8 






58 


















2.2 


5 


68 


26 


S9 


I 













27 


2 










Sio 


2 


I 










76 


15 










Sii 


7 


o 










19 





^ 


20 






Sl2 


7 


I 










7 


I 


90 


9 






Total 


30 


4 


122 


12 


252 


52 


197 


24 


292 


49 


269 


80 


Av. 


2.7 


•4 


40.7 


4.0 


84.0 


17.3 


21.9 


2.7 


36.5 


6.3 


89.7 


26.7 





9A 


9B 


9C 1 


12 A 


12B 


12 


C 




/ 


.y 


/ 


.? 


i 


.y 


/ 


J 


/ 


^ 


/ 


^ 


Si 


57 


13 


64 


13 


148 


50 


104 


74 






108 


82 


S4 


15 


2 


21 


5 






17 


8 


22 


10 






S5 


24 


12 


27 


12 


33 


24 


32 


40 






33 


42 


S6 






II 


7 






80 


28 


83 


31 






S7 






22 


26 


52 


34 


19 


35 










S8 


24 


27 


38 


27 


55 


39 


29 


Zl 










Sio 


98 


41 


103 


41 


118 


48 














Sii 


36 


II 


85 


70 






60 


51 










S12 


105 


II 


125 


II 


251 


45 


164 


149 


172 


149 


190 


160 


Total 


359 


117 


496 


205 


657 


240 


505 


422 


^17 


190 


331 


284 


Av. 


51-3 


16.7 


55-1 


22.8 


109.5 


40.0 


63.1 


52.8 


92.3 


63.3 


1 10.3 


94.0 



say, at the time of completion of the various stages of abstraction 
of the critical numbers, the number of draws from them is on 
the average more than ten times as great as from adjacent non- 
critical numbers. 

The order of abstraction of the various critical numbers is 
also in practically complete harmony with the order of frequency 
of subjects' reactions to them individually. Not only are a sub- 
ject's draws from a lower critical number invariably more nu- 
merous than from a higher one, but his draws from the lower 
non-critical numbers are also fewer than from higher ones. 
Both of these facts tend to give greater emphasis to the lower 
critical numbers and favor the abstraction of critical numbers 
from lower to higher in the order of their magnitude. A com- 
parison of the actual order of abstraction requires that different 
stages of the process be listed separately. This has been done 
in Table IX. 



HIGHER MENTAL PROCESSES IN LEARNING 



A7 



As before the subjects are listed at the left. The critical 
numbers appear at the top of the columns, the various stages be- 
ing segregated as indicated by the capitals. The figures in the 
columns indicate the order in which the critical numbers emerged 
from the various stages of abstraction. Thus the record of Si 
shows that 3 was the first number to emerge from Stage A, 6 
the second, 9 the third, etc. Points where hesitation and other 
inconclusive signs showed the probable occurrence of abstraction 
though no clear indications of the process were observed, are 
marked with asterisks. No overt indications of abstraction were 
observed for the blank points in the table. When a given stage 
in the abstraction of two or more critical numbers was completed 
simultaneously, they were given the same value in the table. 

Table IX 
Order of Abstraction of Critical Numbers. 





3A 


6A 


9A 


12A 




3B 


6B 


9B 


12B 




3C 


6C 


9C 


12C 


Si 


I 


2 


3 


4 




I 


2 


3 


* 




I 


I 


I 


I 


S2 


I 


2 


- 


- 




- 


- 


- 


- 




— 


- 


- 


- 


S3 


I 


- 




- 




I 


- 


- 


- 




I 


- 


- 


- 


S4 


I 


2 


3 


4 




I 


2 


3 


3 




- 


- 


- 


- 


ss 


I 


2 


3 


4 




- 


I 


2 


¥ 




- 


I 


I 


2 


S6 


I 


* 


* 


2 




- 


I 


2 


3 




- 


- 


- 


- 


S7 


2 


I 


* 


3 




- 


I 


2 


* 




I 


- 


2 


- 


S8 


- 


* 


I 


2 




- 


I 


2 


- 




- 


I 


I 


- 


S9 


I 


2 


- 


- 




- 


- 


- 


- 




- 


- 


- 


- 


Sio 


I 


2 


3 


- 




- 


- 


I 


- 




- 


- 


I 


- 


Sii 


I 


2 


3 


4 




- 


I 


2 


- 




- 


- 


- 


- 


S12 


I 


2 


3 


4 




- 


I 


2 


3 




- 


- 


I 


I 



That the correspondence between the frequency of reaction to 
various critical numbers and the order of their abstraction is 
really very close, is shown by the fact that in only one case 
(S7, 3 A) in the entire table did a lower number emerge later 
than a higher one from a given stage of abstraction. There is, 
however, a strong tendency in this group for all critical numbers 
to emerge simultaneously from Stage C. This is due to the fact 
that abstraction has here reached a sufficiently advanced stage to 
permit generalization to become effective. 

h. Effect of Nearness to a Goal. — As already noted the fre- 
quency of reaction to the various critical numbers varies directly 
with their nearness to o, the immediate goal of every trial. It 
is therefore impossible to measure separately the effects of fre- 



48 JOHN C. PETERSON 

quency of repetition and those of the nearness of elements to a 
goal. Logically the relative nearness of elements to a goal re- 
solves itself into two distinct factors : ( i ) the greater ease with 
which the consequences of tentative or hypothetical draws can be 
foreseen from lower than from higher numbers and (2) the 
closer temporal proximity of the goal to reactions toward lower 
than to those toward higher critical numbers. Both of these 
factors were probably operative as will later be pointed out more 
fully. Some evidence of the effects of (i) is found in the per- 
centages of erroneous draws from various non-critical numbers 
as reported in Table VI. In the further presentation of evidence 
at this point no effort will be made to distinguish between the 
effects of these two factors. 

That the critical character of 6 is more easily discovered than 
that of 9, is shown by the number of trials required by 24 sub- 
jects from each of these numbers in discovering the impossibility 
of winning it. The total numbers of trials upon 6b and 9b were 
126 and 194 respectively. Seventeen of these subjects required 
fewer trials upon 6b than upon 9b. Likewise fewer trials were 
required upon 9 than upon 12 beads, both by the group as a 
whole and by a majority of the individual members. But, ow- 
ing to the rapidly increasing effects of previous learning upon the 
higher numbers of the series, the difference here is not so great 
as in the former pair. 

The comparative difficulty of abstraction of various critical 
numbers is further shown in Table VIII. The averages as they 
stand show a marked general increase in the number of reactions 
required to raise successively higher critical numbers out of a 
given stage of abstraction. But this evidence is open to the 
objection that the averages do not all represent the work of 
exactly the same group of subjects. The results are not much 
modified, however, when the work of those subjects whose rec- 
ords do not show complete data for the stages compared, is left 
out of account. In the work of 9 subjects whose records con- 
tain data for both 3A and 6A there is only one case, that of S7, 
in which 3A offers more difficulty than 6A. The average num- 
bers of draws are 3 and 21.9 for 3 A and 6A respectively. Stages 



HIGHER MENTAL PROCESSES IN LEARNING 49 

6A and 9A are both represented in the work of 6 subjects all of 
whom required more draws from 9 in the attainment of 9A than 
from 6 in the attainment of 6 A. The average numbers of trials 
are here 26.7 and 55.8 for 6A and 9A respectively. Data for 
both 9A and 12 A are contained in the records of 6 subjects all 
of whom required more draws for the attainment of 12A than 
of 9A. The average numbers of draws are here 43.5 and 67.7 
for 9 A and 12A respectively. Without going into further de- 
tail it may be said that the results in the B Stages show the same 
marked tendency for higher critical numbers to require many 
more reactions for their abstraction than are required by the 
lower ones. The less marked tendency in this direction in the 
C Stages is clearly due to the onset of effective generalization 
and need not concern us here. 

c. Effect of Speed of Reaction. — The ease with which the 
principle of drawing by "opposites"^^ is discovered appears to 
depend largely upon the speed of reaction of the subject. The 
subjects who reacted rapidly almost invariably discovered the 
"opposite" relation between their draws and those of the ex- 
perimenter early in the game, but those who reacted slowly usu- 
ally became aware of this relation late or not at all. Data re- 
garding the speed of reaction and the number of trials required 
by subjects of Groups I and III are presented in Table X.-^ The 
data for Group I are shown in the upper section of the table, 
those for Group II in the lower section. Since the extent of the 
series is not the same for different subjects, early or late dis- 
covery of the principle of drawing opposites must be determined 
not by the absolute number of trials required for the discovery, 
but by the percentage of all reactions upon Series 1-2 which 
sufficed for the discovery of the principle. These percentages 
are given in the last column of the table. 

22 That is, drawing i when the opponent draws 2, and 2 when the opponent 
draws i. This procedure obviates the necessity of counting the remaining 
beads when their number has once been reduced to a multiple of 3. 

23 Similar data cannot be given for Group II owing to the fact that no 
separate records were kept of the time required for the solution of indi- 
vidual problems of the series. 



50 



JOHN C. PETERSON 



Table X 

Speed of Reaction as Related to Ease of Discovery of the 
Principle of Drawing by "Opposites." 

Per cent of total 
Av. No. of seconds No. of trials re- trials upon Series 
per draw in dis- quired for dis- 1-2 required for 
covering the covery of the the discovery of 

Subjects principle of principle of the principle of 

opposites-* drawing by drawing by 

opposites opposites 



i 


10.25 ( 5.2)25 




I. GO 


ii 


12.28 (12.8) 




I. GO 


iii 


5-94 ( 5-2) 


25 


.89 


iv 


574 ( 6.0) 


25 


.26 


V 


12.76 ( 6.4) 




1. 00 


vi 


2.86 ( 3.4) 


71 


.21 


vii 


5-05 ( 40) 


21 


.06 


viii 


Z-(>7 ( 3-1) 


167 


1. 00 


ix 


4.88 ( 4.5) 


308 


.96 


X 


4.56 ( 6.1) 


77 


•13 


xi 


6.07 (13-5) 




1. 00 


xii 


4-98 (5-0 


70 


.38 


xiii 


2.59 ( 3-1 ) 


86 


.52 


xiv 


3.18 ( 57) 


88 


.24 


I 


6.88' ■(* 5.8) 


32 


•57 


2 


7-49 (5-3) 


^S 


1. 00 


3 


13.13 (11.6) 




1. 00 


4 


3.96 ( 7-2) 




1. 00 


5 


4-8s ( 4-4) 




1. 00 


6 


2.70 (3.1) 


22 


•31 


7 


6.16 ( 5.8) 


35 


.26 


8 


6.08 ( 5.6) 


107 


75 


9 


5-13 ( .S-S) 




1. 00 


10 


3.83 ( 3.9) 


63 


.24 


II 


3-57 ( 4-1) 


III 


.39 


12 


378 ( 31) 


84 


.24 



2* Or until the completion of Series 1-2. 

25 In the parentheses beside the second column the average number of sec- 
onds per draw in the 'first fifty draws of each subject, is given. 

Note the general correspondence between the ease of discovery, 
as indicated by the percentage values in the last column, and 
the speed of reaction as indicated in the second column of the 
table. The average time per draw for subjects who did not dis- 
cover the principle of drawing opposites was 10.34 seconds in 
Group I and 6.77 seconds in Group III. The averages for sub- 
jects who did discover the principle were 4.35 and 5.06 for 
Group I and Group III respectively. By the rank method the 
correlation between the ease of discovery of the principle of op- 
posites and the speed of reaction is .668 for Group I and .516 
for Group III. 



HIGHER MENTAL PROCESSES IN LEARNING 5i 

It might be supposed that these correlations are due to an in- 
crease in the speed of reaction after the discovery of the prin- 
ciple of opposites but before its announcement. That the cor- 
relations were perhaps raised somewhat by this tendency is not 
denied. But the effect of this factor was probably slight since 
the subjects were urged repeatedly to notify the experimenter 
of any new discovery or seemingly relevant new idea at the time 
of its occurrence. Moreover, the records show that subjects 
who reacted slowly at the beginning of the work usually failed 
to discover the principle of drawing opposites, whereas subjects 
who drew rapidly from the beginning were likely to discover the 
principle early in the game. Thus the average time per draw of 
the first fifty draws of Subjects i, ii, v, and xi, who did not dis- 
cover the principle, is 9.47 seconds, whereas the average time of 
the remaining subjects of this group, all of whom discovered the 
principle, is only 4.42 seconds. The corresponding averages 
for subjects of Group III who did and for those who did not 
discover the principle, are 7.20 and 4.60 seconds respectively. 
The correlation between ease of discovery of the principle of op- 
posites and the speed of reaction in the first fifty draws is found 
to be fairly high. For Group I this correlation is .373 and for 
Group III it is .629. 

It may still be argued that the speedy reaction of some sub- 
jects was merely the result of their attention to the relation be- 
tween their own draws and those of the experimenter; but such 
comments of subjects as we have, together with their time rec- 
ords, show that even while attending to the number of beads 
remaining or recalling their success or failure with certain num- 
bers in past trials, these subjects reacted with less deliberation 
than others. As shown by their comments, the work of these 
subjects was characterized by incessant fluctuations of attention 
between relatively isolated elements or by wholly irrelevant re- 
flections. These subjects appeared to react quickly not so much 
because of any marked speed of mental activity as because of a 
lack of those deeper insights into relations and consequences, 
which might have barred hasty reaction. 

In striking contrast with these positive correlations between 



52 JOHN C. PETERSON 

speed of reaction and relative ease of discovery of the principle 
of drawing opposites, are the correlations between speed of re- 
action and facility in finding a solution for the entire series, as 
measured by the number of trials rec[uired for the solution of 
the series. The average speed of reaction in the entire series 
for all subjects of Groups I and II, together with the number of 
trials required for the solution of the series, is shown in Table 
XL'*' 

^ Table XI 

Speed of Reaction as Related to the Number of Trials Required for 
Solution of Series 1-2. 
Average No. of sec- No. of trials required 
Subjects onds per draw in for the solution of 





Series 1-2 


Series 1-2 




10.25 


50 


ii 


12.28 


35 


iii 


5-52 


28 


iv 


6.89 


96 


V 


12.76 


62 


!vi 


3-19 


331 


vii 


4.00 


371 


viii 


2.^7 


167 


ix 


4.60 


315 


X 


4-33 


597 


xi 


6.07 


53 


xii 


6.54 


184 


xiii 


3.68 


166 


xiv 


2.77 


367 


A 


12.22 


48 


B 


7.00 


62 


C 


5.61 


85 


D 


11.86 


112 


E 


6.70 


220 


F 


11.67 


300 


G 


4.68 


322 


H 


7.00 


389 


I 


4.86 


617 


J 


3-91 


823 



The data for Group I are given in the upper portion of the 
table and those for Group II in the lower portion. The corre- 
lations between speed of reaction and the number of trials re- 
quired for the mastery of the series are — .748 for Group I and 
— .686 for Group II, when the speed of reaction is stated in 

26 Similar data are not available for Group III owing to the failure of 
most of the subjects of the group to obtain a solution for the series while 
solving the limited number of problems presented to them. 



HIGHER MENTAL PROCESSES IN LEARNING 53 

terms of the average time per draw in the entire series. When 
the speed of reaction is determined on the basis of the first fifty 
draws of each subject, the correlation is ■ — .264 for Group I. 
Thus intelHgence as determined by the ease of solution of Series 
1-2, does not affect the speed of reaction so much at the begin- 
ning of the series as later when there is a greater accumulation 
of experience which may serve to inhibit hasty reaction. Speed 
of reaction appears then in the double role of an effect of the 
depth of insight into the conditions of the problem and as a 
factor in determining the direction of analysis. 

2. Explicitness and Extent of Analysis 

In the analysis of puzzles Ruger reports a "wide variation of 
felt clearness from extremely vague to perfectly clear. This 
range of felt clearness," he finds, "is matched by differences in 
results." Our results show a similar variation in the explicitness 
of analysis from cases in which an element of a situation is 
barely recognized in passing, to those in which clear verbal form- 
ulations are accompanied by ability to recognize and control the 
element in question under novel conditions. Numerous cases 
were observed where analysis was complete enough to be effec- 
tive for manipulation but not clear enough to be put into words. 
A great many of the perceptual solutions were of this character. 
The analysis of the lower numbers practically never reached the 
explicitness of a clear verbal formulation until generalizations 
made upon higher numbers were applied to them. Occasionally 
a subject would declare that he knew how to win a given num- 
ber but could not express it in words. One subject, after win- 
ning from 20 beads twice in succession declared that she believed 
she could do it again though she could not say how. She then 
proceeded to win two more trials in succession without difficulty. 
Another subject, having just won twice in succession from 10 
beads, was asked whether she had any new ideas. She replied : 
"I can't state the idea in words but I have it in motor terms." 

Ruger mentions several different sorts of analysis which af- 
fected only a part of the situation. A very indefinite type of 
partial analysis he describes as "picking out the portion of the 



54 JOHN C. PETERSON 

puzzle to be attacked. In many cases," he says, "there was a 
mere spatial analysis, 'locus analysis' without involving any per- 
ception of the mechanical necessities concerned." This locus 
analysis resulted in the confining of random movements to a por- 
tion of the puzzle. This is very similar to a common experience 
of our subjects many of whom quickly discovered that the cru- 
cial point of the trial lay in the early draws, and busied them- 
selves, accordingly, with the task of determining what the first 
one or two draws must be. This often resulted in some im- 
patience with the requirement that every trial must continue un- 
til the number of beads was reduced to o, and necessitated a few 
brief departures from the rule. Often the point conceived of as 
crucial was in the lower and sometimes in the intermediate num- 
bers rather than in the higher ones, as the following examples 
show: 

"I wonder if you often draw 2 at the beginning as I do, merely 
to hurry and reduce the number of beads" (xiv, 23b). ^^ 

"No matter what number you are working with you can win 
if you can get it to 4 or 5 beads. The problem isn't with the 
larger numbers but with the few at the end" (xii, lib). 

"It seems to revolve about the start" (xii, i6b). 

"Well, I would eliminate the beads until I got my first turn on 
4 or 5 (xii reacting to the problem of 59 beads). 

"When I win, the crucial draw is the second, or not later than 
the second" (xiv, 25b). 

A slightly more definite type of analysis is described by Ruger 
as follows : "An important form of partial analysis noticed was 
that of a single step in the process while the other steps were 
attained only by random movement. This single step was often 
the final one. The solution would come accidentally but the sub- 
ject would notice the last step. In subsequent trials he would 
know what to do if he chanced to get to that step but not how 
to get there." The folowing comments from subjects indicate 
this type of partial analysis : 

2^ The Roman numerals in parentheses indicate the subject whose com- 
ment is quoted and the numbers following show the particular problem in 
the series, during the solution of which the observation occurred. 



HIGHER MENTAL PROCESSES IN LEARNING 55 

"I can't beat when we get to 6. I never have beaten you on 
6" (xiii, I2b). 

"I can't win this; there are 6 left" (ix, lib). 

"I could win if it would come out ii there" (ix, 14b). 

"I know I can't win unless I can make it 13 there . , . after 
you have drawn" (vi, 25b). 

"Some of the first moves I win and others I lose. Now on 5 
if I have the first move, I win" (vii, 22b). 

"I was trying to get it so there would be 7 beads left on my 
draw" (viii, 12b). 

These partial analyses, especially when concerned with the 
higher numbers in a series, depended largely upon the possession 
of some general scheme by means of which attention could be 
freed from most of the details of the series and concentrated 
upon some one point. Such schemes usually took the form of 
drawing uniformly either i or 2 beads throughout the trial, and 
generally resulted quickly in some sort of appreciation of the 
principle of drawing by opposites. This principle is one of the 
best examples found here of what Ruger called "schematic an- 
alysis." A partial insight into the significance of multiples of 3 
sometimes served as a schema of this sort. One subject, for 
example, expressed a schematic view of the line of approach to 
a solution of Series 1-2 as follows: "You draw 2 when I draw 
I, and I when I draw 2, so that we reduce it by 3's each time; 
that is, we are drawing them out by multiples of 3" (vii, i8b). 
From this comment it might be thought that the subject had ef- 
fected a complete analysis of the series, at least up to and in- 
cluding 18 beads; but it required two additional trials to bring 
him to realize that he could not win 18 beads, and many more 
trials to complete the series. During the next trial he observed ; 
"We are drawing right down on multiples of 3" ; but here again 
he failed to realize the significance of multiples of 3, for it re- 
quired still another trial to convince him that he could not win 
18 beads, and his conclusion was not based on multiples of 3 at 
all. "I can't win from 6," he said, "so since this is a multiple of 
6, I can't win it." Of the problem presented by 18 beads the 
subject had probably made a total analysis in the sense that it 



56 JOHN C. PETERSON 

"reached all the elementary steps or movements." That it was 
far from a complete analysis of the entire series is evident from 
the fact that it required still 45 trials upon 19, 20, 21, and 22 
beads for the development of sufficient insight to permit the sub- 
ject to give a solution for the entire series. At this stage of the 
process, however, analysis proceeded far less than formerly by 
frequency of repetition, nearness to a goal, etc., and far more 
by means of ideas and principles formulated in early problems 
and now applied to new problems and verified with a minimum 
of repetition. In other words, generalization, which has played 
a role of increasing importance since the earliest stages of analy- 
sis, has here become the dominant factor. 

3. Time Relations of Manipidation and Analysis 

In describing the time relations of ideational analysis and 
motor variations Ruger says : "These two types of variation, 
acts of analysis and motor responses, may be quite varied, espe- 
cially in their time relations. At the one extreme is the motor 
variation which, perhaps, brings success but which runs its course 
unnoticed. At the other extreme the analysis may occur first 
and only after a considerable interval be followed by the motor 
response. "^^ This wide variation in the time relations of analy- 
sis and manipulation is very characteristic of the work of our 
subjects, except that in the first series we find little ideational 
analysis preceding manipulation. Our procedure was not of 
course such as to encourage attempts at analysis prior to manipu- 
lation in the early problems of the series, though we believe that 
such efforts were seldom discouraged except by their failure to 
bring results. Such analysis as occurred early in the series was 
usually of a perceptual nature, as already explained, and was ac- 
companied and checked by manipulation of the beads. The ex- 
tent to which the method of trial and error became operative in 
the intermediate and higher numbers of the series, is indicated 
roughly by the percentage of erroneous draws. These per- 
centages for some of the subjects in the first six problems of 
the series are given in table V. In approximately the first two- 

-s Op. cit., p, 12. 



HIGHER MENTAL PROCESSES IN LEARNING 57 

thirds of the trials of a subject upon each of the higher numbers 
the erroneous first draws usually amounted to about 50 per cent 
of all of the draws from the number in question. Often, indeed, 
half of the first draws of all trials in a number of successive 
problems, were erroneous. Thus Subject viii, in working from 
7 to 14 beads in the first series, made 86 errors of this sort out 
of a possible 175. In the 7 problems from 8 to 14 beads in- 
clusive the possible errors for this subject were 156 and his 
actual errors were 86. Subject i required 38 trials in working 
over the numbers from 7 to 14 inclusive in the first series; i.e., 
there were 38 possibilities of erroneous first draws. Although 
this subject was one of the most successful of the lot and made 
special effort at analysis before manipulation, he made 21 errors 
in the 38 draws. His comments upon his methods of procedure 
are of interest here : "I'm doing this by the trial and error 
method until I get it down to 4 or 5 where I can handle it." 
Later he says : "I find that any attempt to analyze this series is 
apt to inhibit action; at least it doesn't conduce to a solution." 
Repeated comments of this nature were made by many of the 
subjects. 

It often happened, while drawing through a trial which had 
been given up as lost, that success would be accidentally achieved. 
Frequently the subject would attempt to recall how it was done. 
Often he succeeded, but even if he failed, the effort was not en- 
tirely lost, for the arousal of attention to the existence of ne- 
glected possibilities resulted in a change of attitude and some- 
times in a fruitful examination of his assumptions. Numerous 
occurrences of accidental success without ability to recall the 
successful variation were noted. The following case is repre- 
sentative : 

Subject ix had experienced particular difficulty with 11 beads, 
Series 1-2, and finally concluded that he could not win if the 
experimenter drew i early in the trial. He was persuaded to 
continue, however, and finally happened to take 2 at every draw 
and so won the trial. His comment was : "I just fell on to that 
by trial and error procedure. It was accident; I don't believe I 
could do it again." Some time afterwards he met with the same 



S8 JOHN C. PETERSON 

difficulty when drawing from 14 beads. After winning with 
considerable difficulty, he said: "I don't know how I did that." 
Later on with 17 beads he repeated his old error seemingly none 
the wiser for his two accidental successes. 

A striking example of how accidental successes may be seized 
upon and made the basis for further progress is found in the 
reactions of Subject vi who, having made good progress up to 
25 beads in Series 1-2, became stranded and required 156 trials 
to win this number twice in succession. This subject soon dis- 
covered that when his initial draw was i and the experimenter 
drew 2, he could win by taking i at every draw throughout the 
trial. But when the experimenter took i at the first draw, the 
subject was helpless, though he could have won by merely chang- 
ing his draws to 2 from this point to the end of the trial. Some- 
how the latter procedure escaped his attention, although he fol- 
lowed it once by accident near the beginning of the plateau, so 
that with 57 opportunities to win by this procedure he succeeded 
only twice. After his 155th trial upon 25 beads, when the suc- 
cessful variation occurred for the second time, he said : "That's 
another way I win — certainly I could win that way if I won the 
other way !" When questioned as to whether he had known from 
the beginning of the trial that he would win, he replied : "Not 
until the third to the last draw. In fact, I started drawing 2 
without any intention of continuing to draw 2 on down through 
the trial. Then I later decided to continue (because, as subse- 
quently brought out in a comment, he did not remember having 
tried that before) as I started. ... It didn't occur to me until 
after I had finished, that it didn't make any difference whether 
I got that one extra bead first or last, if you drew i each time." 

The last sentence throws some light on the nature of the diffi- 
culty experienced by him with this combination. If the subject 
and the experimenter drew i bead each at the first draw, there 
would be 23 beads remaining. Now, if the subject took i bead 
at his second draw and the draws proceeded by opposites after 
that, there would be 2 beads left for him at the last draw; that 
is, "i extra bead." The subject's reactions clearly show that 
some such assumption as, that he must provide for that "extra 



HIGHER MENTAL PROCESSES IN LEARNING 5Q 

bead" at his first opportunity, was made; for of the 55 errors 
which he made in drawing i when he should have taken 2, ioriy 
consisted in drawing i from 23 beads. The other 15 errors of 
this sort can be accounted for largely on the basis of his efforts 
to reach certain lower numbers which he was gradually learning 
to regard as significant. This false assumption explains why the 
subject made 40 errors in 57 draws from 23 beads. The case 
shows how an accidental success may serve to clear away false 
assumptions and so contribute indirectly as well as directly to a 
more speedy solution of the problem. 

It is worthy of note that the accidental successes in the fore- 
going cases, which occurred before a fair acquaintance with the 
series had been formed, either failed to attract attention or could 
not be recalled, whereas an accidental success occurring later in 
the series was quickly seized upon and applied to the solution of 
the series. This was quite a common occurrence in the work of 
a majority of our subjects. 

Having seen the futility of early attempts at analysis without 
manipulation, and the varying degrees in which subjects profited 
by accidental success at different stages of their mastery of a 
series, we turn briefly to the growing capacity for ideational 
analysis, which usually began to be effective towards the end of 
Series 1-2 and became increasingly apparent as the experiment 
progressed. The first successful attempts at ideational analysis 
of more than perhaps one step in the series usually consisted in 
a return to lower numbers which were regarded as related to the 
present situation, with an attempt to determine their forgotten 
status, or in a return to the very beginning of the series to re- 
cover orientation. Reasoning about these lower numbers would 
invariably take the form of ideational manipulation mentioned 
earlier in the discussion. Usually some numbers remembered as 
critical served to facilitate the determination of the status of 
other numbers. An example will show the character of this 
sort of analysis more clearly : 

After gaining a fair acquaintance with Series 1-2 in 371 trials 
upon the first 17 problems of the series, Subject vii took an 
excursion back to the beginning to check up and get his bearings. 



6o JOHN C. PETERSON 

"Some numbers," he said, "the first (draw) wins and some it 
loses. Now at 5 beads if I have the first move, I win; at 7 I 
lose — I take 2 and yon take 2 and I lose. From 7 if I take 2, 
that leaves you 5 and you can take 2 and win. If I take i and 
leave 6 — if I have 7 and take i, I win; If I have 9, I lose. If 
I have 9 and take i, I lose the game; if I have 9 and take 2, I 
lose, therefore I lose on 9 always. If I have 8 and take i, the 
other fellow wins; if I take 2, — O yes! / see why it is that I 
zvin 8, because I leave 6. I can win 5 by taking 2 and I can win 
7 by taking i ; 6 I can never win. If I have 8 and take 2, I 
always win. I can win 5 if I take 2 ; 6 I can never win, 7 I 
can win if I take i, and 8 I can win if I take 2 ; 9 I can never 
win — 6 and 9 I can never win; 10 if I take 2, I lose; if I take i, 
I can always win. 11 if I take i, I lose; if I take 2, I can always 
win. 12, if I take 2, you can take i and leave 9; if I take i, — I 
can't win it. 5 I win and 6 I lose — on even numbers I take 
away i: On 11 if I take 2 — 12 I lose because you can leave 
either 9 or 8. 6, 9, 12 I can't win. I can't win any multiple of 3. 
On other multiples of 5' — I can't carry it out. Let's go on." 

Here the subject was asked how he would draw from 59 beads. 
"An odd number," he said. "This will be merely a guess. I 
should judge that since it is an odd number, I could win. I 
would try to leave it odd all the way. I would take 2 and the 
other fellow couldn't win if I watched my draws. I think he 
couldn't win if I watched my draws. I think he couldn't win 
multiples of 3' — I would reduce him to midtiples of ^!" 

Even after the solution of an entire series most of the sub- 
jects showed but slight tendency towards or capacity for antici- 
patory analysis in the attack upon new series. Twelve of the 
fourteen subjects in Group I began the attack upon Series 1-3 
in exactly the same manner as that upon the first series; that is, 
they began to draw without delay and only after some time at- 
tempted to formulate a statement for the series. The effect of 
their Avork upon the preceding series was very noticeable, how- 
ever, in the readiness with which some subjects took note of the 
fact that 4 could not be won, and in their alertness for other 
critical numbers. 



HIGHER MENTAL PROCESSES IN LEARNING 6i 

Subject i said at the beginning of Series 1-3: "I am going 
to try to reduce this trial and error to a minimum. With i or 
2 (i.e., in Series 1-2) the smallest combination was 3; i, 2, or 3 
is a more difficult proposition. I imagine I could sit down and 
work it out." He drew 2 and the experimenter took the remain- 
ing 3. "Oh, I forgot," he said "that there were three possi- 
bilities. I will try all the possibilities now and see if you can 
get me." From this point manipulation and analysis progressed 
together. This failure of one of the most successful subjects, 
certainly the one who was most inclined towards anticipatory 
analysis, shows the difficulty of such analysis in the early stages 
of acquaintance with the elements of the problem. His progress 
had been, however, very rapid in the first series so that time 
had not permitted a very thorough stamping in of the elements, 
and he had altogether failed to notice some of the important 
elements of the series. 

When presented with Series 1-3, Subject xiv began to draw 
after slight hesitation but said immediately after the first trial : 
"Now I think I can get this right off. This time you are going 
to let me win every fourth one; that is, on the fourth you won't 
let me win and on the fifth you will let me win, provided I start 
with I. On 6 I should have to start with 2, and with 3 on 7. 
(And on the 8th?) I wouldn't win 8." This subject quickly 
solved two more series of this order without a draw. 

This exceptional case may well be due to the fact that Subject 
xiv made a more thorough analysis of Series 1-2 than was ef- 
fected by any other subject. Also by a great deal of shifting of 
attention from one set of elements to another during 367 trials 
occupying 8173 seconds, as compared with an average of 204 
trials in 4943 seconds for all subjects of Group I, he had prob- 
ably succeeded in stamping in these elements more thoroughly 
than had been possible for most of the subjects. With less than 
half as many trials as xiv required for Series 1-2, Subject i 
solved in the neighborhood of thirty series of problems and gave 
satisfactory generalizations for all problems included in the 
experiment. It is clear, therefore, that the insight shown here 
b}^ Subject xiv cannot be regarded as a case of ideational analy- 



I 



62 JOHN C. PETERSON 

sis without a fair acquaintance with the elements of the situation. 
It required the solution of only one or two additional series 
of problems to give this control of all series of the first order 
(i.e., all series in which L is equal to i) to nearly all of the sub- 
jects, and all succeeded in getting a general solution after only a 
few more series. But no subject solved Series 2-3 without 
manipulation of the beads. The solution of fewer series was re- 
quired, however, for the mastery of the second order of series 
than for the first order. Likewise the mastery of all series of 
the third order required actual work on fewer series than that 
of either of the preceding orders, and usually resulted in the 
mastery of all series in which the numbers between H and L 
might be drawn. This transfer of power from series to series 
was due largely to the easy recognition of certain elements of 
the new series, which were identical with familiar elements of 
old ones, thus obviating in part the necessity for new analytic 
activity. The transfer of power was further facilitated by gen- 
eralizations upon familiar elements, which were easily applicable 
to new series. These factors of familiarization and generaliza- 
tion became more prominent as the work progressed, but in 
series where some new elements were introduced there was still 
a ncessity for genuine analytic learning. Much of this analysis 
continued to follow or accompany manipulation, but a gradually 
increasing proportion was performed in ideational terms, largely 
by the sort of ideational trial and error already described. Thus 
with increasing acquaintance with the materials we find a con- 
tinual receding of activity from overt trial-and-error manipula- 
tion to a very similar sort of process carried on in ideational 
terms, which in turn gives way to general ideas that have evolved 
gradually by the bringing together of similar elements in asso- 
ciation with general symbols. In the later series all three of these 
factors were almost constantly in operation. 

4. Summary 
After the range of perceptual solutions was passed various 
elements of the problematic situation were gradually abstracted 
and associated with verbal symbols. The type or combination 



HIGHER MENTAL PROCESSES IN LEARNING 63 

of types of elements so abstracted differed largely for different 
subjects and in different periods of the work of the same sub- 
ject. The type of elements abstracted was shown to vary in ac- 
cordance with the speed of reaction. Frequency of reaction to 
various elements and their nearness to a goal, were shown to be 
closely correlated with the order of their abstraction. Large 
differences in the degree of explicitness of analysis were ob- 
served, and there was found to be a somewhat gradual develop- 
ment from vague to clear and explicit states of analysis. The 
extent of analysis varied with different subjects and at different 
stages of the learning process, but analysis usually occurred first 
in isolated spots and spread from these to other portions of the 
series. The time relations of ideational analysis and manipula- 
tion varied greatly. In general, manipulation preceded analysis 
in the first few series, and persisted throughout the greater por- 
tion of the learning process as an important method of procedure. 
Gradually, however, as acquaintance with the elements of the 
situation grew, overt trial and error was replaced by a very 
similar sort of ideational manipulation which, in turn, tended to 
give way to general ideas. 

E. Generalization 
In the preceding section attention was directed principally to 
the abstraction of the elements of the problems presented for so- 
lution. As was shown in several instances, however, such ele- 
ments as are abstracted do not remain in their early state of rela- 
tive isolation during the entire course of analysis, but tend to 
combine into higher units which become associated with appro- 
priate symbols and take on general meanings. This process of 
generalization and the resulting general ideas will be the subject 
of discussion in the present section. 

I. Relative Absence of Generalisation in the Perceptual Stage 

Mention has already been made of the apparent lack of gener- 
alization during the period of perceptual solutions. Practically 
no comments were made during this period, which would seem 
to indicate any attempts at generalization. Moreover, when a 



64 JOHN C. PETERSON 

subject did finally attempt to generalize, those numbers which 
are well within the range of perceptual control were rarely taken 
into account, though they had been more often repeated than any 
of the higher numbers. Only two of the fourteen subjects of 
Group I noticed, with sufficient clearness to mention the fact, 
that 3 is a critical number, until attention was directed to it by 
their final generalizations which were based upon higher num- 
bers. Some of the subjects were very much surprised that this 
fact had so long escaped their attention. The first important 
landmark mentioned by eleven of the subjects of this group was 
6, and this was generally mentioned not earlier than the last 
trial on g beads, sometimes not until much later. 

But this failure to recognize the status of lower numbers in 
the abstract does not preclude recognition of the critical or non- 
critical character of the concrete situations for which they stand. 
In fact, the situation which is represented by the critial number 
3, was almost invariably the first to be reognized in the concrete 
as critical. This fact is shown in Table VIII where the realiza- 
tion of the impossibility of winning 3 is shown to have been the 
first step taken by ten of the twelve subjects of Group III in the 
abstraction of critical numbers. Table IX, however, shows that 
3 was explicitly mentioned as a critical number before 6 was so 
mentioned, by only 2 of the twelve subjects. The failure of sub- 
jects to utilize numbers below 6 along with higher numbers as a 
basis for generalization, is not, therefore, due to the difificulty of 
recognizing their status in the concrete but rather to the ease of 
such recognition, which made it unnecessary to associate these 
situations with verbal symbols. 

2. Development of the Concept of the Critical Number 
Though the concept of the critical number seems to be a very 
simple affair, its development required the expenditure of a con- 
siderable amount of time and effort on the part of the subject. 
To facilitate the description of this development we may divide 
it into the following seven stages : 

Stage A. — ^The first appearance in explicit form of the criti- 
cal-number idea was at the point where the subject discovered 



HIGHER MENTAL PROCESSES IN LEARNING 65 

that he could not win 6 beads. But this was straightway forgot- 
ten in most cases, or at any rate failed to function in slightly 
new situations. Similarly, the impossibility of winning 9, 12, 
etc. was later discovered without a full realization of the signi- 
ficance of these numbers at the time. This first discovery of the 
impossibility of winning a number constitutes the first stage in 
the growth of the concept of the critical number. Critical num- 
bers in this stage stand out in comparative isolation. 

Stage B. — Since the subject was required to draw first in every 
trial, it was impossible to compel him to draw from a given 
critical number in any trial in which the inital number of beads 
was only i greater than the crtical number in question. There- 
fore during the solution of the next higher number in the series 
the recently discovered critical number was usually forgotten. 
Its re-discovery after more or less delay constitutes the second 
stage in the development of the concept. Here the individual 
critical number is recognized as an element of all problems of 
the series; e.g., 6 is recognized as critical regardless of what 
may have been the initial number of beads presented in the trial. 

Stage C. — At this stage the relationship between two or more 
critical numbers in Stage B, or perhaps in Stages A and B, was 
discovered; as, 'T cannot win 9 because you can always reduce 
me to 6"; or, 'T cannot win 12 because I lose 6, and 12 is com- 
posed of two 6's," etc. Here the critical numbers are not only 
recognized as critical in all problems of the series, but they are 
associated with each other in more or less definite relations. 

Stage D. — The fourth stage consists in the discovery of the 
fact that the subject cannot win any multiple of 3. All critical 
numbers are recognized as critical and all are associated with a 
common symbol. 

Stage E. — The foregoing stages all deal with the critical num- 
bers as related to the subject's recognition of the possibility of 
success. A somewhat similar development may be observed in 
his utilization of the critical-number idea as a means of control 
in the achievement of success. Thus it often happened that a 
subject discovered the possibility of winning by reducing the 
number of beads to 6 and so forcing the experimenter to draw 



66 



JOHN C. PETERSON 



from it, long" before learning that the same procedure applied to 
the higher critical numbers would bring the same results. In 
this stage one particular number is recognized as critical for the 
one zi'ho draws first. 



Stage F.- 



-In Stage F two or more critical numbers are recog- 



nized as having- this broader significance. 
Stage G. 



-Here all critical numbers are recognized as critical 
for the one who draws first. 

It must not be supposed that learning- progressed with perfect 
regularity from stage to stage in exactly the foregoing- order, 
or that all of the stages were distinguishable in the progress of 
every subject. They do, however, occur separately in the ma- 
jority of cases, and in some instances in the exact order named. 
The following comments of subjects will serve to illustrate this 
development. They are all taken from Series 1-2. The stage of 
development of the concept, as inferred from each comment, is 
indicated in parentheses. The portion of the series in which the 
comment occurred is in each case indicated at the left. 12b, 12, 
for example, signifies that the first comment occurred during 
the twelfth trial on 12 beads. 

Subject vii 
12b, 12: "You alwa3^s manage to leave me 6." (B) 
" 26: "I can't win 12 because you can always reduce it to 6; 12 is two 

6's." (C) 
15b, 11: "You reduce it to 6 every time." (B) 
" 22: "You always manage to get to 6; I don't know how I am going 

to prevent you." (B) 
" 24: "There are 12 left; I can't do much with that." (B) 
" 24: "There are 9 left; I haven't been able to win any 9's." (B) 
i8b, 8: "Evidently this is one that you don't expect me to take because 

you throw it into the 6's." (C) 
" 9: "Your idea is to balance my moves so as to make it 6; and 9 is 

just as good for you." (B) 
" IS : "You draw 2 when I draw i and i when I draw 2, so that we 
reduce it by 3's ; i.e., we are drawing them out by multiples of 

3." (C) 

" 16: "We are drawing right down the multiples of 3." (C) 

" 17: "This is a multiple of 6; I can't win from 6 so I can't win it." (C) 

2ob, I : "The subject draws two and is asked why. "I have no reason," 

he replies. E then draws 2 and the subject takes i, explaining: 

"I draw that way to keep it on multiples of 5 rather than on 

multiples of 3, and so to avoid 12, 9, and 6." (C) 



HIGHER MENTAL PROCESSES IN LEARNING 



67 



" 2: After drawing 2 S says: ''I can't win with 18." (B) (E?) 
22b, 9: "If I have 8 beads and take i, the other fellow wins. If I take 
2 — Oh yes, I see why I win: because I leave 6. (E) ... I can't 
win .any multiple of 3. ... I can't carry it out; let us go on." 
(D) The subject is here asked how he would draw from 59 
beads. After some irrelevant speculations he concludes : "I 
would take 2 and the other fellow couldn't win if I watched my 
draws. I think I would reduce him to multiples of 3." (G) 

Subject xiii 

"It depends on which draws first." (E?) 

"I can't win when we get to 6. I never have beaten on 6." (B) 

"I couldn't w^in 6 ; I don't believe I can win 12. I have been trying 
to see how to prevent you from leaving me 6. Double it and it 
is the same. It would be the same on all multiples of 6." (C) 

"I can't beat you on this ; it goes by 3's. / have forgotten whether 
I beat you on 3. 6 is the first I fell down on, isn't it? I don't 
know about 9." (D) 

"I can't beat you on 16 — Oh no, it was 18 I lost. (B) If it is 20 
and I draw first, I win; but if you draw first, I lose. Let me 
see, I didn't beat you on 18. If you draw first on 18, I win." (F) 

"I don't believe I can beat 18, 12, or 9, or 6. It must be multiples 
of 6 I can't win." (C) (D?) 

"When I get 26, I see that by drawing 2 first I can give you 24, 
and I can therefore win." (F) 

"I don't believe I can win. You make me draw on 24 no matter 
how I move. It must be multiples of 3, i.e., 3, 6, 9, 12, 15, 18, 
etc. That's what it is! (D) It depends on who draws first. 
There are certain numbers that the one who draws first can win 
and others that his opponent can win. (G) ... After your first 
draw, i.e., after you get it to that number, draw opposites." The 
subject is here asked how he would draw from 59 beads and 
replies : "I would take i and, provided you take 2 every time — 
let me see, there may be another element there. I must draw i 
first on the odd numbers which I can win and 2 on the even 
numbers that I can win. But I am not certain yet." 
28b, 3 : "It looks like it is even numbers that I draw i on ; and on odd 

numbers I draw 2 perhaps." 
29b, i: "That's the way it is. 3's I can't win, and multiples of 3. (D) 
4 and 5 I win ; 7 and 8 I win ; 10 and 12 I win, etc. So on even 
numbers I draw i and on odd numbers I draw 2, and the oppo- 
site after the first draw." 
31b, i: "My scheme didn't work!" 
" 2 : "I have missed something there." 

" 3 : "It is a question of making you draw first on the multiples of 3. 
I didn't draw 2 here so that was wrong (i.e., his idea that he 
must draw 2 from odd numbers, etc.). It is a question of what 
to draw so as to make you draw first on a multiple of 3." (G) 



lob. 


6 


12b, 


30 




36 


15b, 


6 


20b, 


2 


24b, 


2 


26b, 


2 


27b, 


2 



68 JOHN C. PETERSON 

Subject xii 
gb, o: "The first I couldn't win was 6, wasn't it? This is 9." 
" 8: "Is it possible that with 6 or multiples of 6, such as 9 and 12, 
one can't get any results? that is, on all multiples of 3? (D) 
... In all cases of multiples of 3 I am going to assume that I 
can't get any results." 
12b, 9: "I am positive I can't get this. I always begin with 6 (B). 
. . . I came back to the original hypothesis and thought of 6 as 
a multiple of 3." (D) S is here asked how he would draw from 
59 beads. His reply shows that he has failed utterly to realize 
the significance of his idea of multiples of 3 for the achieve- 
ment of success: "Well, I would eliminate the beads until I 
got my first draw on 4 or 5." 
14b, 6: "When there are 6 and it is my turn to draw, you can always 
win." (B) The subject does not realize yet that 9 and 12 are 
also critical numbers. 

"You can always make up the deficiency so as to take away an 
even number and leave 6." (B) 

"I am going to try out that hypothesis again, that any number 
which is a multiple of 3 precludes any possibility of success for 
me." (D) 

"Whenever a number is a multiple of 3, it is useless for me to 
try it." (D) 

"When it gets to 9, I lose. (B). In this case I am working on the 
first 7 out of 16." 

"It seems to revolve about the start." 

"No matter how I move you can always bring me to 9. I lose 
12 too." (B) 

"I'll bet a cow I can't win 18." S is here asked how he would 
draw from 59 beads, and replies : "I couldn't get it." 
"I lose multiples of 3." (D). When asked how he would draw 
from 61 beads, he says: "I would draw 2. (Why 2?) To 
get to 59 and bridge over 60. (E?) 
22b, I : "Well, I lose and I thought I had it cinched." The subject later 
stated that he took 2 beads at the first draw in this trial in order 
to "bridge over" 21. 
22b, 2 : "I see my mistake. I took 2 and you 2 leaving 18, another multi- 
ple of 3-" (D) 
24b, o: "I'll pass it up." (D) 
25b, I : "That was foolish. I took 2 and allowed you to reduce it 21, a 

multiple of 3, and I lose." (D) 
27b, o: "I'll pass it up." (D) 

30b, o: "I'll pass it up." (D). The subject was asked how he would 
draw from 82 beads. He replied: "I would take i. No, I 
would take 2, because that would throw me off the multiple of 
3 again." (E?) From 67 beads he said he would take i "to 
avoid 66, a multiple of 3." (D). This subject did not advance 



14b, 


9 


15b, 


I 


15b, 


7 


i6b. 


4 


(( 


16 


*' 


37 


i8b, 





2ib, 


I 



9b, 


5 


nb, 





" 


4 


11 


6 



HIGHER MENTAL PROCESSES IN LEARNING 6g 

beyond Stage D in Series i-2, although he seemed at the point 
of making the larger generalizations at several times. 

Subject i 

"I cannot beat you at g. No matter how I draw you can take 
such a number as to reduce me to 6 which I can't win." (B) 

"Now if I knew what my winning numbers were for 7, 8, 9, and 
4, and 5, I think I could win." 

"Did I win 6 before? I don't think so." (B) 

"You reduce it to 6 no matter how I draw, and 6 is impossi- 
ble." (B) 
" 7: "The first party can't solve 11 if his opponent reduces the num- 
ber to 6 (E). I have it. . . . If I can prevent you from re- 
ducing the number to 6, I can win." 
12b, 3 : "It reduces itself to this : Can I prevent you from reducing it 
to 6? You can leave 6 irrespective of what my move is. Twelve 
is composed of two 6's." (C) 
13b, I : "Now I want to prevent you from getting 6 or 12, so I must 
necessarily start with 2 (E?). (S draws 2 and E i) Nine 
beads. There you have got me because I take i and you 2 and 
leave me 6, or I take 2 and you i and leave me 6." (C) 
" 2: "I'll take the other chance and start with i. No there is no use 
because I would be leaving you 12. Well, as far as I can see 
my only chance is to start with 2, but even then you can reduce 
me to 6 if I am not mistaken. 
" 3: "It seems to me that I lose. If I take i, you have 12 and win; 
if I take 2, you can bring it down to 9 and hence to 6 (C). If 
you can reduce me to 6 or 12 no matter how I draw, why I 
lose, that is certain. Let's see if I can beat you on 6. . . . 
No I can't beat you on 6, that is evident. I have an idea that 
it is multiples of 3 you win." (D) When asked how he would 
draw from 59 beads, he replies : "56 is the nearest multiple of 
3 to 59. If you could reduce it to 57, you would have me. There 
again I can't move so as to prevent you. If I took i, you would 
take I and leave me 57. If I took 2 — Oh yes ! If I moved 2 
and followed it out consistently, I think I would have you 
beaten." (F, G?) The subject here stated that he had a glimpse 
of the right procedure at 11 beads, but that some distraction or 
other caused him to lose sight of it for a time. 

Subject viii 
lib, 20: "When there are 4 and it is E's draw, I can get it. My problem 

is to get it reduced to 4 when it is his draw. (E?) 
12b, 18 : "The idea strikes me that this cannot be gotten, and also that it 

is a multiple of 6 that cannot be beaten." (C) 
13b, 18: "First I wanted to get 4 beads with my draw; I later found that 

I would have to work not from 4 but from 5. . . . Later I 



70 JOHN C. PETERSON 

changed it to 7. I thought I could work down to 4 from 7, 
and then I began to work from 9. I then found that I would 
have to begin to check off from the total; that is, if there are 
13 beads, I draw i and then count 12. . . . My object was to get 
g beads when it mas my draw." (E?) 
" ?,2'- "I counted back to 11 and worked it out from 11. Here / tried 
to get my opponent down to 6 instead of myself. (E) 

14b, 12 : "I forgot that I was trying to keep from getting 6, and conse- 
quently for a good while I was trying to get 6, and not being 
able to win caused me a good deal of confusion. I have had 
this mixed up for some time." 

15b, 7: "I am now working on 9 instead of 6." (F) 

i6b, 4: "I think that the point of the game is that the one who wins 
has to have the other one draw on some multiple of 3." (G) 

The foregoing comments of subjects are merely milestones in 
their progress through Series 1-2, yet if the reader will refer to 
the position in the series where each comment belongs, he will 
appreciate the extreme slowness with which the simple meanings 
often developed and the lack of uniformity in their attachment 
to various elements of the series. It has already been shown 
that the lower critical numbers are usually wholly neglected for 
a time after their first discovery and that they are later re-dis- 
covered in about the same order as that of their first discovery. 
In general, each new acquisition of meaning — each advance to a 
higher stage in the foregoing classification^ — began with those 
critical numbers above the range of purely perceptual solutions, 
which had been most often repeated, and spread to the higher 
critical numbers much in the order of their frequency of occur- 
rence in the work of the subject. Thus 6 was the first critical 
number, usually, to be re-discovered, and also the first to be 
definitely associated with a higher critical number. So also 6 
and the immediately following critical numbers were usually the 
elements between which the multiple-of-three relationship was 
first apprehended though higher critical numbers were at the 
time known to the subject. Likewise 6 was usually the first 
critical number to advance to Stage E, and the higher critical 
numbers followed roughly in the order of their magnitude. It 
has already been shown that the reactions of any subject to a 
lower critical number were invariably more frequent than his 
reactions to a higher one. Thus in the elevation of critical num- 



HIGHER MENTAL PROCESSES IN LEARNING 71 

bers to higher levels of meaning frequency of repetition and 
nearness to a goal appear to be important factors, just as in the 
preceding section they were shown to be factors in the direc- 
tion of attention to the critical numbers. 

In the elevation of critical numbers to higher levels of mean- 
ing there was, however, considerable deviation from the order 
of their frequency of repetition. Thus 12 was often taken up 
into Stage C upon its first discovery long before 9 was re- 
discovered. This was clearly due to the simple multiple-relation 
existing between 12 and 6, by which the associations already 
built up about the latter were made to attach immediately to the 
former. ^'^ This spread of meaning through the agency of pre- 
viously acquired associations might well be expected since the 
elements of our problems were not only familiar to the subjects, 
but were also closely related in past experience. But this calling 
up of familiar concepts by association is itself perhaps dependent 
in large measure on the frequency of their previous repetition in 
association with some element of the present situation. These 
irregularities in the process of learning did not greatly alter the 
general character of the learner's progress in the early stages of 
learning. The generalization of the formula for Series 1-2 into 
a suitable formula for all series of the first order^*^ required from 
I to 6 further series for its completion; and the generalization 
of the latter formula so as to make it cover all continuous series"^ 
required from 2 to 10 additional series. It is therefore evident, 
notwithstanding frequent rapid advances by means of familiar 
concepts which were brought in by means of association, that 
the development of meaning was a rather slow and fluctuating 
affair, presenting some striking parallels to the formation of 
sensori-motor co-ordinations. 

Another fact of interest, which is exemplified in some of the 
comments listed above, is the fleeting and unstable character o£ 
meanings in the early stages of their development. Individual 
critical numbers and generalizations based upon them were given 

29 For example, see the foregoing comments under Subject i, 12b, 3; 
Subject vii, 12b, 26; and Subject xiii, 12b, 36. 

30 Series in which the value of L is i. 

31 Series in which all numbers between L and H may be drawn. 



72 JOHN C. PETERSON 

to slipping- out of mind and to various distortions of meaning, 
which clearly reveal the weakness or utter lack of association 
with other elements of the situation. Thus Subject i, while 
working upon his seventh trial with 1 1 beads, appears to have 
had a fleeting insight into the significance of critical numbers as 
a means of actual control. Several other subjects spoke of a 
similar tendency for promising ideas to vanish during the at- 
tempt to formulate or apply them. Sometimes a critical number 
would be recognized and remembered well enough, but with an 
inversion of meaning somewhat comparable to the reversals of 
perspective in ambiguous drawings. Thus Subject xiv (lib, 
29) said: "Whenever I win, I always have to leave 3 beads; 
that is, there must be 3 beads before I take my last draw." This 
sort of inversion of meaning often occurred shortly before the 
appearance of Stage E in the development of the concept of the 
critical number. After trying for some time at this point to 
get to draw from 6 beads subject viii said: "I forgot I was try- 
ing to keep from getting 6, and consequently for a good while 
I was trying to get 6." Another instance is found in the com- 
ments of Subject i quoted above (13b, i). Similarly the idea 
of the significance of multiples of 3 often dawned and was lost, 
later to reappear, perhaps several times, before finally being ap- 
plied in a thoroughgoing way with a full realization of its sig- 
nificance. Examples of the early instability of this idea are 
found in the comments listed above as follows: vii, i8b-22b; xii, 
9b-3ob, and xiii, I5b-27b. In tracing the further consolidation 
of the various elements of the concept of the critical number 
through succeeding series of problems, it is found that they are 
often thrown out of co-ordination by the appearance of some new 
element which requires adjustment and further generalization, 
or by the recall of some old hypothesis which has not been satis- 
factorily disposed of. 

The interference of old and erroneous hypotheses with the 
formation and stability of correct generalizations, is worthy of 
further attention. The comments of Subject xiii, quoted above 
(27b, 2), show how the idea that odd and even numbers were 
possessed of special significance, came in to confuse him after 



HIGHER MENTAL PROCESSES IN LEARNING 73 

he had made a perfectly good generalization. This subject had 
played somewhat with the idea of odd and even numbers much 
earlier in the series, but during the greater portion of the work 
he seems entirely to have dismissed the idea, making no mention 
of it at all. The very common interference of erroneous hy- 
potheses and irrelevant ideas which have once received attention 
without being definitely settled, will be further illustrated in the 
following paragraphs on trial and error in generalization. 

3. Random Hypotheses 
Having observed the slow growth of meanings and their in- 
stability in the early stages of development, we may turn briefly 
to the manner in which the elements of a situation were brought 
together and combined into adequate generalizations. It rarely 
happened in the early members of a group of series that the 
essential elements were directly sought out and held in mind 
without attempts at generalization until sufficient data were at 
hand to insure correct inference at once. This type of procedure 
was followed apparently by one of our subjects in Series 1-2, 
but was not carried out consistently in all later series. The work 
of all other subjects in the first series showed some loss or omis- 
sion of essential data with consequent erroneous generalizations. 
These random hypotheses were often numerous and far afield. 
The rate of progress depended very much upon the thoroughness 
with which they were followed up and tested. Because of the 
unusual fullness of his comments the work of Subject xiv will 
be taken to illustrate the random efforts at generalization in the 
first series. The place in the series at which each comment oc- 
curred will be indicated in the manner already familiar to the 
reader. 

9b, 3 : "You can always win if you draw tlie same as I." 
lib, 20: "If, with an odd number, you take an even number, I can always 

beat you if I work it right." 
" 36: "Whenever I get a combination by which I can win, thereafter I 

must reverse when you reverse in order to win." 
15b, 6 : "I win some odds and some evens." 

" 7: "I draw another conclusion, that my principle is not to follow 
you." 



74 




« 


8 


" 


9 


I7b, 


I 


(( 


9 


i8b, 


2 


2ib, 


7 


22b, 


2 


25b, 


I 


" 


12 



13: 



31b, 12: 
32b i: 



34b, 21 : 

36b, I : 

37b, 4: 

40b, I : 



40b, 5 : 

41b, 4: 

42b, I : 

43b, 4: 



JOHN C. PETERSON 

"Neither is it to draw opposite from you." 

"Neither is it to alternate with you." 

"I can always beat you on even numbers ; I don't think I can 
beat you on any odds." 

"There must always be 3 beads when you draw last." 

"As long as you draw opposite to me I can't win 18." 

"You win odd numbers by taking the opposite of my draw." 

"The principle of alternation will not work with the even numbers." 

"Even and odd numbers is not the criterion." 

"There must be some relation between the total number of beads 
and the number of draws." 

"When I win, the critical draw is the second or no later." For 
some time after this the subject's attention was absorbed in the 
number of beads he obtained in each trial. 

"I haven't kept track of the numbers I won and those I lost." 

"To win an odd number of beads I draw an odd number." Here 
again the subject's attention is absorbed for some time in the 
number of draws and of beads obtained in each trial. 

"2 from 34 leaves 32. Now if I draw i always and you 2, I would 
leave even numbers." ( !) 

"It seems that I can't win every third time." 

"Of course I know you are going to take 2 all the time and I'll 
take I, but I don't know how it will come out." 

"I will beat you but I can't say how with any degree of certainty. 
I'll beat you twice and then you will beat me once. I notice that 
the numbers upon vi^hich you beat me are always divisible by 3." 
The subject is asked how he would draw from 59 beads and re- 
plies that he cannot tell. He then figures slowly but correctly to 
59, after which he is asked how he would draw from 1004 beads. 
"I couldn't figure that up," he said, "I may win even numbers 
and you odd numbers. No, I won 27- It may be numbers that 
are divisible by 2. No, I don't get it. . . . It might be that you 
allow me to beat you on all numbers that are divisible by a 
certain number, but it isn't by 2 and it isn't by 3 or 4. ... I 
have never figured out this drawing business ; I do it wholly ac- 
cording to my feeling. But I watch the way you draw on your 
second draw and then take the opposite. ... I am not sure what 
is the criterion, whether it is your first two draws or my first 
two." 

"I ought to have noticed that 3-matter long before because 1 
noticed so often that we drew i and 2." 

"I knew I could beat; I had 3 beads after my second draw." 

"I can't beat you on 42, it is divisible by 3. I know what you 
can beat me on but I don't know why." 

"They go in pairs. I can beat you on 43 if I take 2 the first 
time (He loses). You haven't arbitrarily made up your mind 
not to let me beat you the first time, have you?" 



HIGHER MENTAL PROCESSES IN LEARNING 



75 



" 8 : "When I win I guess I must draw 2 the first time ; but I can't win 
that way, because I've tried it and failed. . . . But if you hap- 
pen to be drawing the opposite of what I draw the second time, 
I'll beat you. . . . I'll take 2 and then you'll let me beat you." 
" g: "Well, I thought all I had to do to beat you the second time was 
to reverse what I did the first. I see, evidently I have to start 
out with the same number in order to beat you on a given num- 
ber." 
" 10 : "Three from 43 leaves 40. All right, I have it. I have an even 

number; I'll take 2." 
" 12: "I had 15 draws — Oh yes, that is all right. I was thinking I 
could get out by leaving an odd number of beads." 
44b, i: The subject takes 2 and wins. "That confirms my rule," he says. 
4Sb, o : "I can't win 45." 

46b, I : "I can win 46. I'll have to start with 2. Yes, I must or I can't 
win, I believe." 
" 3 : S draws i and wins. "On the numbers on which I can beat 
you," he says, "I have to start out with i." 
47b, I : "I can beat you on this. . . . Well, that rule didn't work (i.e., 
that S must always draw i first to win). Well, maybe the rule 
is that one time I draw i in order to win and the next time I 
draw 2 to win." 
48b, : "Did I win 47 by drawing i ? No, I won it by taking 2. Then I 

can't win 48, of course." 
49b, I : "I'll draw here on the hypothesis that on the first one after the 
number I lose, I can win by taking i, and on the second after it 
I must start with 2 to win." 
. " 2: "It worked. Now this time I'll start with 2 (S loses). No, that 
isn't right." 
"Now I'll start with 2 and beat you." Here the subject is taken 
back to the beginning of the series. 
(S takes 2 and loses) "Now that's funny." 
"I'll take I to start with." 

S takes 2 and loses. "I've got to start out with i on 7," he ob- 
serves, "and there is a lot of other numbers en which I have 
found that I must start with i." 
"I'll try 2 just to see. No, I'll try i." 

"Why isn't the rule that on the even numbers I've got to start 
with 2?" 
"Yes, that's the rule." 

"Oh, I can't beat you on 9, and on 10 I've got to start with 2." 
"Oh! I mustn't take 2 the first time; I must take i." 
"I believe my hypothesis was right about there being first the i- 
draw and then the 2-draw. On 4 I had to draw — I forget just 
how I did draw on 4. On 10 I drew i and on 11 I took 2. 1 
think I must take 2 on 11." 
12b, o: "I can't win 12; it is a multiple of 3." 



50b, 


I 


7b, 


I 


" 


2 




3 


8b, 


I 


<( 


3 


« 


4 


9b, 


I 


lob, 


2 


lib, 


2 



76 JOHN C. PETERSON 

15b, o: "I can't get it." 
i6b, o: "Now I'll take i." 

17b, I : "I'll draw 2." The subject was then asked how he would draw 
from 59 beads. "Why," he said, "it goes this way. I draw I 
first, then you draw 2 and I keep on with i. Then I beat you. 
The next timt^^ I start with 2 and you start with i, and I keep 
on with 2 and win. But I can't tell how a given nuiriber must 
be drawn on. I just know that it goes along i and 2. What is 
this number? (17) Well, on 17 I had to draw 2 didn't I? Well, 
I should guess that on 59 I should start with 2, and then you 
would draw i and I would continue with 2. (Why?) The only 
reason I say that is that it is just before 60, and 60 is a number 
that you beat me on just as you do on 18. Seventeen is just be- 
fore 18 and 59 is just before 60." In reply to a question as to 
how he would draw from 43 beads the subject said: "I would 
draw I on 43, because on 42 I couldn't beat you, and with the 
numbers following those on which I can't beat you I must al- 
ways start out on i." 
The random making of hypotheses upon the basis of momen- 
tary suggestions is here too evident to require further comment. 
Examples might be produced ahuost indefinitely from Series 1-2. 
A few examples from higher series and from the efforts of sub- 
jects to get a generalization of universal application for all series 
will perhaps be worth while. 

The following evidences of random hypothesis-making are 
taken from the record of Subject viii. In attacking Series 2-3 
this subject fluctuated a good deal among a number of hypothe- 
ses, as is shown in his comments : 

7b, 2 : "I think now that I will win on the basis of 7, i.e., I Avill try al- 
ways to draw so that there will be 7 left to draw from." 

8b, 2 : "It appears that I can win on a basis of 5." 

9b, 2 : "It appears that 7 doesn't work." 

" 3 : "It doesn't work. I forgot about the i." 

" 6: "It will work on a basis of 6 as well as of 5, i.e., on multiples of 
both 3 and 2, I believe. But I am not certain." 
32 Note the possibility of confusion in the phrase, "next time." It may 
mean the next trial upon the same number or it may mean the next higher 
number. These two meanings fluctuate in the mind of the subject, owing 
perhaps to the ambiguous wording. The same ambiguous wording appears in 
the comments of this subject at 47b, i (p. 94), though here it is clear that 
the subject has the correct idea in mind. But it is the wording rather than 
the idea, that functions in the immediately following problems (47b, i to 
7b, 3) in such a manner as to produce a number of errors. Several other 
instances were found of ambiguous wording which led other subjects into 
confusion and error. 



HIGHER MENTAL PROCESSES IN LEARNING 77 

15b, 3 : "I am a little in doubt about the multiple of 3 and 2. It doesn't 

seem to work." 
20b, 3 : "Twenty can't be worked. I didn't think of that multiple until I 

had worked for a long time. It is a multiple of 5." 
2ib, I : "Sometimes the multiple of 3 works and sometimes it doesn't. 

Either a multiple of 2 or a multiple of 3." 
2ib, s : "At ifirst I thought I could work on the basis of 2 or 3, but one 

of the multiples of 3 didn't come out right. I almost think now 

that I made a mistake and that I can't work on this basis." 
26b, 2: "That hypothesis couldn't have been right because 12 is a multiple 

of both 2 and 3 (S won 12 repeatedly), so I have given that up." 

The subject was tormented by these and numerous other hy- 
potheses until he had solved all of the numbers of this series from 
6 to 5 1 and then repeated the process up to 29 beads. 

Perhaps the most striking exhibition of this sort of random 
effort was found in the attempts of subjects to formulate a gen- 
eral solution for all series. Here it was usually necessary to give 
them a list of the critical numbers of all discontinuous series in 
order to refresh their memories. ^^ All of the subjects had made 
some discoveries, as for example, that all multiples of L + H 
are critical numbers. The generality of some of these earlier 
formulatonis was readily recognized if not already known. But 
some were not universal in their application, and here is where 
the effort began to be more sporadic. The subject would usu- 
ally begin by restating one of his old generalizations or by for- 
mulating a new one on the basis of one or two problems, and 
then go from series to series testing it out until a series was dis- 
covered to which it would not apply. Upon finding a refractory 
series he would stop and attempt to modify his view to suit the 
case in hand and then pass on to test it out upon other series. 
Often the subject would have to modify his generalization for 
every group of series which were in any marked degree different 
from the preceding ones, sometimes discarding the notion only 
to return to it later in his fumbling efforts to find a generaliza- 
tion which would be applicable to all series. Some subjects 
failed after working for hours, and most of the others succeeded 
in getting a general solution only by a tedious "fitting on" of 

^3 Series in which only L and H may be drawn. For a list of these simi- 
lar to that presented to the subjects see page 7. 



78 JOHN C. PETERSON 

numerous hypotheses suggested by individual series and often 
wholly at variance with conditions which had been met repeated- 
ly in other series. A better exhibition of random effort could 
not have been given by an animal in a problem box. 

Nor is it to be supposed that in the application of generaliza- 
tions to new situations we have an entire escape from trial and 
error procedure. Numerous errors and omissions of application 
point to the random character of efforts to apply generalizations. 
Certain hindrances to application and some sources of error are 
fairly apparent in the transition from the continuous to the dis- 
continuous series of problems and in passing from the first to 
the second series of the latter group. 

Three of the more important elements of their previous gen- 
eralizations which subjects brought over into Series i or 3 are 
(i) the critical-number concept, (2) the idea of the serial rela- 
tion of the critical numbers of the series, and (3) the knowl- 
edge of the fact that the common difference between successive 
terms of the various series of critical numbers is equal to L -)- H. 
No difficulty was experienced in applying (i) and (2) directly 
to Series i or 3. The trouble with the application of (3) arises 
from the fact that a new series of critical numbers is created by 
the restriction of the draws to L and H. The first term of this 
new series is 2 and the common difference is L -|- H as before. 
This new series of critical numbers should have given but little 
difficulty if the subjects had stuck to the old generalization for 
primary critical numbers and looked for an explanation for the 
secondary^^ critical numbers only. But there were too many 
pitfalls in the way. In the first place, the first critical number 
encountered was a secondary one. This refractory case imme- 
diately cast doubt upon the applicability of the old formula and 
lessened its chances of later consideration. Again, the secon- 
dary critical numbers fall into line with the primary ones in such 
a manner as to form a continuous series with a common dif- 

L-fH 

ference of . This is, of course, a coincidence peculiar to 

2 

2* See above, p. 8 for definition of these terms. 



HIGHER MENTAL PROCESSES IN LEARNING 79 

this series and not common to all. Finally, the common differ- 

L + H , 
ence of is 2, in this particular case, making all even num- 

2 

bers critical. Thus the mind of the subject is led unawares away 
from the general principle built up in previous series and di- 
rectly into the rut of the old concepts of even and odd numbers. 
The power of these superficial relations and the familiar con- 
cepts aroused by them to divert the attention of subjects from 
the application of generalizations formed in previous series, is 
evident from the fact that twelve of the thirteen subjects who 
solved this series, stated their final formulae in terms of even 
numbers, though five of them had played with the old formula 
while working upon the early problems of the series. 

Some interesting errors of application were made by Subject 
viii in passing from Series i or 3 to Series i or 5. After having 
generalized correctly for Series i or 3 he began his attack upon 
the following series correctly enough by saying : "I would think 
right away that 6 is the lowest critical number and that all multi- 
ples of 6 are critical." But he quickly got off the track and began 
to believe that all multiples of 3 were critical. Later he sup- 
posed that the critical numbers were all multiples of 9, but finally 
changed his conjecture to multiples of 12. After obtaining a 
solution for this series he stated that the errors had arisen from 
his efforts to adapt the generalization of the preceding series — 
i.e., that "All multiples of 2 are critical" — to the present one. 
Examining the former series to find the basis of the distribution 
of critical numbers, he observed that 2 lies midway between i 
and 3, the L- and H-draws respectively, and straightway in- 
ferred that multiples of 3 were critical in Series i or 5 because 3 
lies midway between the L- and the H-draw in this series. Find- 
ing this hypothesis to be incorrect he tried the sum of all the 
numbers between i and 5, i.e., 9, because it was observed that 
"2 constitutes all the numbers between i and 3. This failing, 
he "thought that a critical number had to be a multiple of all of 
the numbers between i and 5, i.e., a multiple of 12. 

One of these errors was repeated by this subject in passing 
from Series 2 or 6 to Series 2 or 10. In generalizing upon the 



8o JOHN C. PETERSON 

former series he said : "Since about the beginning of the last 
series I have felt that it must go by multiples of 4, because 4 is 
intermediate between 2 and 6 and is a factor of both." Imme- 
diately afterward, when presented with Series 2 or 10, the sub- 
ject said without hesitation: "I should say the critical numbers 
are multiples of 6 and numbers which are i above those multi- 
ples; that is, I add 2 to 10 making 12 which I know is a critical 
number. Then I reduce to 6 — No, I didn't, did I? — as I had, 
in the other case, reduced 8 to 4." 

In generalizing upon Series 3 or 9 Subject ii said: "The criti- 
cal numbers are multiples of the difference between the high and 
the low draw." When presented with the next series, he said: 
"Immediately I take the difference between 4 and 12, which is 8. 
This formulation applies very well to all series in which H = 3L, 
but to no others. Although the subject did not explicitly say so, 
he undoubtedly carried this generalization over into Series i or 
4 and Series i or 6 much to the detriment of his progress. While 
working upon 9 beads in Series i or 4 he decided that 3 was a 
critical number and said : "I feel that I am on the edge of some- 
thing here ... 3 I lose. It reduces to 3." Shortly afterwards 
he added : "I've been mistaken there. I lose 3 and 5. No, I 
win 3." After thinking this matter over for two minutes he 
continued : "I believe I can win 3, 6, and 9." A little later, 
after a short pause to determine how to draw from 9 beads, the 
subject said : "Now that's funny. I can win 9." The idea that 
multiples of 3 are critical numbers in this series seems finally to 
have been discarded at this point, but the generalization upon 
which it rested, — i.e., that the difference between successive criti- 
cal numbers is equal to H — L, — persisted into the next series 
and seems to have been the cause of considerable difficulty there. 
The idea, carried over from Series i or 3, i or 5, and i or y, 
that even numbers are critical, is also much in evidence in the 
records of Series i or 4 and i or 6. The ill effect of these gen- 
eralizations from earlier series upon the progress of the subject 
through Series i or 4 and i or 6, is evident in his records. 
Though in the speed of progress throughout the entire experi- 
ment this subject ranks second, he is tenth in rank in Series i 



HIGHER MENTAL PROCESSES IN LEARNING 8i 

or 4 and i or 6 combined. The other nine of the ten subjects 
who solved all of the series devoted only lo per cent of all their 
trials to these two series, whereas Subject ii devoted 32 per cent 
of all his trials in the entire experiment to the solution of them. 
It appears, therefore, that approximately two-thirds of the diffi- 
culty encountered by Subject ii in these two series, was due to the 
interference of certain generalizations carried over from pre- 
ceding series. ^° 

Many other examples of the difficulties attending the applica- 
tion of generalizations could be given but the principal types are 
perhaps sufficiently illustrated in the foregoing cases. The omis- 
sions and errors of application occurring in the records of all 
subjects are numerous enough to warrant the statement that, 
under the conditions of our experiment at least, the application 
of generalizations depends very much upon trial and error pro- 
cedure. It appears that any incompleteness of analysis, either in 
the old situation from which the generalization is evolved or in 
the new situation to which it is to be applied, is likely to render 
application difficult and perhaps erroneous. There is no apparent 
reason why difficulties of this nature should arise from the con- 
ditions of our experiment more readily than from the practical 
situations of life. 

4. Summary 

Explicit generalization did not often occur until the number 
of beads presented was high enough to preclude the possibility of 
direct, perceptual foresight of the consequences of all possible 
draws, i.e., until the numbers presented were high enough to 
place a premium on the use of symbols in their solution. Num- 

35 It is of course possible that other factors contributed to this retardation 
of progress. Fatigue may have affected the progress in Series i or 4. This 
series was completed shortly before noon when the experiment had been in 
progress approximately three hours. The subject stated that he was be- 
coming somewhat fatigued though not until after much difficulty had been 
encountered. However, the greater portion of the effort Vv'as expended upon 
Series i or 6 where fatigue conld not have been a factor. This series was 
begun after an intermission of one and a half hours, when the subject de- 
clared that he was thoroughly rested. 



82 JOHN C. PETERSON 

bers lower than this were seldom taken into account even in later 
attempts at generalization. 

Several fairly distinct stages in the development of the criti- 
cal-number concept were found. Each advance to a higher level 
of meaning began usually with that critical number, above the 
range of perceptual solution, v^^hich had been reacted to most 
frequently, and affected the higher critical numbers largely in 
the order of the frequency of their repetition. This is also the 
order of their numerical and temporal nearness to the goal, i.e., 
the end of the trial. In the early stages of their development 
the meanings involved in the concept of the critical number v\rere 
found to be extremely unstable in character. The stability of 
these meanings gradually increased with continued reaction to 
the situations from which they were evolving. 

The selection of the essential elements of a series and their 
combination into adequate generalizations was effected mainly 
by means of trial inferences, or random hypotheses, which were 
made upon the basis of only a few cases — often only one — and 
tested out more or less persistently before final acceptance or re- 
jection by the subject. 

In the application of generalizations to situations which are 
in part new, much difficulty was experienced, and a consider- 
able amount of random modification often occurred before a 
successful adjustment to the requirements of the new situation 
could be made. False analogies arising from the observation 
of superficial relations often resulted in confusion and error. 
Sometimes the solution of new problems was much retarded by 
the attempted application of inadequate or irrelevant generaliza- 
tions formulated from the elements of earlier series. 

F. Transfer 
The term, transfer^ is used by Ruger broadly "to include the 
effect of any given experience on any subsequent one whether 
the effect results directly or by means of an idea, whether the 
transfer is one of method or of material, or of motor processes, 
and whether it is positive or negative."^" The term will be used 

36 Op. Cit., p. 85. 



HIGHER MENTAL PROCESSES IN LEARNING 83 

in the same broad sense in this discussion. This usage is becom- 
ing fairly common. Ruger's defense of it need not detain us. 

I. Degree of Transfer 

Time and other limitations have not permitted such experi- 
mental evaluation of the various problems as would be required 
for a highly accurate measurement of the transfer of the effects 
of learning from problem to problem. However, the facilitating 
and, sometimes, the inhibiting effects of earlier upon later efforts, 
are great enough to give a fair insight into the causes and con- 
ditions of transfer without a very precise determination of its 
amount. A rough estimate of the relative difficulty of various 
problems and series of problems can be made from some of the 
experimental data and other facts. 

Within a given series the higher problems are undoubtedly 
more difficult than the lower ones. If the subject's draws were 
determined by pure chance, two successive winnings would re- 
quire 8 trials upon 5 beads, 32 trials upon 8 beads, 128 upon 
II beads, and 512 upon 14 beads. This increasing difficulty is 
sufficient to conceal the effects of transfer in many instances, 
especially in the lower numbers of the series. Though no ex- 
perimental evaluation of the various problems of any series was 
attempted, the records of Group III when compared with those 
of Group II A show that the solution of 14 beads was consider- 
ably more difficult than that of lower numbers of the series. 
Group HA began with 4 beads and followed the same procedure 
as Group I. Group III followed the same procedure but began 
with 14 beads instead of 4. The number of trials required by 
each member of this group for the solution of 14 together with 
some data for comparison from Group IIA, are given in Table 
XII. 

The subjects of Group III required a greater number of trials 
for the solution of 14 beads than were required by those of 
Group IIA for the solution of the first seven problems of the 
series, and over half as many as were required by the latter 
group for the solution of the first ten problems. Assuming that 
the two groups were possessed of equal ability to solve problems 



84 



JOHN C. PETERSON 







Table XII 












Group IIA 










[otal No. i 


of trials 


Total No. of trials 










required \ 


tor the 


required for so- 


No. of trials 


required for 


solution 


of all 


lution of all 


solution 


of I. 


4 




problems 


from 


problems from 


beads 






4 to 10 in- 


4 to 13 in- 










inclusive 


inclusive 


Group IIA 


Group 


III 


Subj. A 


32 


48 





Subj 


. I 


8 


" B 


39 


49 


5 




2 


II 


" C 


25 


68 


2 




3 


15 


" D 


56 


88 


4 




4 


27 


" E 


34 


106 


4 




5 


29 


" F 


59 


84 


4 




6 


35 


" G 


36 


112 


22 




7 


42 


" H 


74 


97 


25 




8 


50 


" I 


47 


159 


2 




9 


60 


" J 


39 


92 


32 




10 
II 
12 


63 
180 


Total 


441 


903 


100 






606 


Average 


44-1 


90.3 


lO.O 






50.6 



of this sort, the transfer from an average of 90.3 trials upon the 
first ten problems to the eleventh problem of the series was equal 
to 40.6 trials, or to 80.2 per cent of the average number of trials 
required for the solution of 14 beads when presented as the initial 
problem. Further profitable discussion of transfer from prob- 
lem to problem within a series must await a careful determina- 
tion of the difficulty of these problems. 

From logical analysis it would seem that all series of the same 
order are of approximately equal difficulty and that series of 
higher orders are somewhat more difficult than those of lower 
orders. ^^ Some experimental evidence of the substantial equality 



2^ In Series 1-2 there are just four combinations of draws by which sub- 
jects can win non-critical numbers. Upon problems in which the initial 
number of beads is greater by i than a multiple of H -f- L the subject can 
win half of the trials by taking i bead at every draw. The other half he 
can win by taking i at his first draw and 2 at every draw thereafter through- 
out the trial. He can win one half of the trials upon problems in which the 
initial number is less by i than a multiple of H -f- L, by taking 2 beads at 
every draw. The other half of these problems he can win by taking 2 at 
his first draw and i thereafter throughout the trial. If L and H be substi- 



HIGHER MENTAL PROCESSES IN LEARNING 85 

of difficulty of Series 1-2 and Series 1-3, is found in the records 
of Groups IIA and IIB, which are summarized in Table XIII. ^® 
The subjects are listed at the left in the first column. The num- 
ber of problems solved by each subject in each of the series and 
the number of trials and amount of time required for the solu- 
tion of the series, are given in the succeeding columns. 

tuted for i and 2, these four combinations of draws are effective for the 
solution of some problems in every series of the first order. But for all 
problems in which the initial number is removed from a multiple of H -]- L 
by an interval greater than i, a new combination of draws must be made 
by varying the first draw. In Series 1-3 a new combination is required for 
only those problems in which the initial number of beads is an odd multiple 
of 2; i.e., for one-third of the non-critical numbers of the series. In Series 
1-4 it is required for half and in Series 1-5 for three-fifths of the non- 
critical numbers, etc. 

Moreover, the discovery of the successful combinations of draws is not 
favored at so early a point in the higher as in the lower series, owing to 
the magnitude of the H-draws which effect a reduction of small numbers 
to o before a sufficient number of repetitions has been made to impress the 
uniformity of response upon the mind of the subject. For the same reason 
the early discovery of the principle of drawing by opposites is less likely 
in the higher than in the lower series. Again, because of the longer inter- 
val between critical numbers in the higher series, a greater number of prob- 
lems must be solved in these series than in lower ones in order to reveal a 
given number of critical numbers upon which to base and with which to 
test generalizations. 

On the other hand, each critical number in the higher series is more firmly 
fixed in mind than those of the lower series by the more frequent reaction 
of subjects to it before the next higher critical number is presented. This 
would favor the more ready utilization of critical numbers in the higher 
series for purposes of generalization and so facilitate the solution of these 
series. This higher degree of learning is the only factor revealed by analysis 
which clearly favors the more rapid solution of higher than of lower series 
of the same order. How important these various factors are cannot be 
stated definitely without further experimentation. It does not seem un- 
reasonable, however, on the basis of this analysis, to assume that the higher 
series are at least as difficult as lower series of the same order. 

Critical numbers are relatively more numerous in higher than in lower 
orders of series, and their relations to one another are more complex. It 
is practically certain, therefore, that a series of a higher order is more diffi- 
cult than one of a lower order. 

38 The subjects of these groups were members of a class in educational 
psychology, whose class work had been observed by the writer for from 
three to seven months. The division into groups was made upon the basis 
of reasoning ability as judged by the writer. 



86 



JOHN C. PETERSON 



Table XIII 
Group IIA 



Series 1-2 



Subj. 
A 
B 
C 
D 
E 
F 
G 
H 
I 

7 



No. 
Problems 
10 
15 
IS 
16 
21 

74 
27 
76 
81 
162 



Trials 
48 
62 

85 
112 
220 
300 
332 
389 
617 
823 



Time 
(in seconds) 

1833 

1555 

1840 

4210 

6450 
19133 

7440 
13663 
15988 
18572 



No. 
Problems 
12 



Series 1-3 

Trials 
37 
18 

13 

24 

15 
31 
o 
16 
17 
524 



Total 

Average 



321 

32.1 



2196 
219.6 



81136 
8113.6 



71 
7.1 



158 
15.8 



Time 

1130 

230 

248 

685 

130 

1545 

o 

312 

162 

10260 



Total 


497 


2988 


90684 


153 


695 


14702 


Average 


49-7 
No. 


298.8 
Series i 


9068.4 

Group IIB 

-3 

Time 


15-3 

No. 


69.5 
Series 1-2 


1470.2 


Subj. 


Problems 


Trials 


(in seconds) 


Problems 


Trials 


Time 


a 


16 


48 


2545 


9 


14 


300 


b 


16 


50 


2062 


12 


29 


763 


c 


19 


56 


4545 


9 


IS 


407 


d 


16 


69 


2640 


4 


10 


135 


e 


17 


95 


2090 


3 


7 


70 


f 


23 


179 


9585 


I 


2 


120 


g 


38 


210 


8455 


6 


14 


270 


h 


26 


337 


10131 


12 


31 


532 


i 


52 


376 


18648 


3 


6 


150 


J 


98 


776 


20435 


12 


30 


1054 



3801 
380.1 



The average number of trials per subject in Series 1-2 is 
298.8, P.E., 51.9. In Series 1-3 the average number of trials per 
subject is 219.6, P.E., 46.3. The difference between these aver- 
ages is 79.2, P.E., 69.5. In so far, therefore, as the calculation of 
unreliability means anything when based upon so few and so 
variable data as those at our command, these data support the 
view that there is no substantial difference in the difficulty of dif- 
ferent series of the same order. 



HIGHER MENTAL PROCESSES IN LEARNING 87 

In terms of the number of trials required to solve the series, 
the transfer from Series 1-2 to Series 1-3 is yG.y per cent. From 
Series 1-3 to Series 1-2 it is 92.8 per cent. The lower percentage 
of transfer in the former case is due almost wholly to the be- 
havior of Subject J. The insight of this subject into the first 
series at the time of her successful generalization was unusually 
superficial and, contrary to custom, five days were permitted to 
pass between her completion of the first series and her attack 
upon the second. If we leave her record out of account, the de- 
gree of transfer for the group becomes 92.1 per cent. The trans- 
fer from either series to the other is here so great as completely 
to overshadow such differences as may exist in the difficulty of 
the two series. 

The degree of transfer from series to series within the vari- 
ous orders of continuous series is shown in Table XIV. The 
percentages of transfer are based on the supposition that all 
series in any given order are equal in difficulty. The series are 
listed at the left of the table in the usual order. Individual sub- 
jects of the group are indicated by the Roman numerals at the 
top. Though not all of the series were specifically solved by all 
subjects the general solutions given for all series of an order 
were such as to indicate that all higher series of the order could 
be solved without manipulation of the beads. We. have there- 
fore figured the average number of trials per series just as if 
every series had been specifically solved by every subject. 

The degree of transfer may be further traced from lower to 
higher orders of series. As already stated, the higher orders 
are probably more difficult than the lower ones, but in the absence 
of experimental data upon this point we shall assume that all 
orders are of equal difficulty and calculate the percentages of 
transfer upon that basis. The transfer from lower to higher 
orders, as found in the records of the thirteen subjects who 
solved all of the continuous series, is shown in Table XV. 

Further evidence of transfer is found in the constant decrease, 
through successively higher orders, in the number of series re- 
quired to be solved for the mastery of an entire order. The 
average numbers of series solved by actual manipulation of the 



JOHN C. PETERSON 



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HIGHER MENTAL PROCESSES IN LEARNING 89 

Table XV 
Transfer from lower to higher Orders of Series 

Per cent of trials Percentage of 
Order Ave. No. of trials saved as compared transfer from pre- 





per 


subject 


vi^ith 


first order 


ceding order 


First 




230.8 








Second 




103-5 




55-3 


55-3 


Third 




21.6 




90.6 


79.2 


Fourth 




54 




977 


75-1 



beads in the various orders of continuous series were 3.38, 2.46, 
1.08, and .23 for the first, second, third, and fourth orders re- 
spectively. 

Owing to various irregularities which make their compa.ra- 
bility uncertain, no attempt will be made to determine the degree 
of transfer in the discontinuous series. That much positive 
transfer occurred can readily be seen by referring to the data 
upon these series in Table III. 

2. Conditions of Transfer 

No attempt will be made to enumerate all of the conditions and 
sources of transfer but a few of the more important ones will 
be mentioned and discussed briefly. 

a. Objectively Identical Elements. — In order to solve any of 
the higher problems of a series it is necessary to make all of the 
moves appropriate to the solution of all lower non-critical num- 
bers of the series. All of the lower numbers of a series are 
therefore common elements of the higher problems of the series. 
Thus in Series 1-2 the numbers 4, 5, 6, 7, 8, etc. must all be 
solved in the process of solving the higher numbers of the series. 
That the solution of these lower numbers contributes to the 
mastery of higher ones is shown by the results reported in Table 
XII. Also the effect of the presence of common critical num- 
bers is shown by the fact that subjects soon learned the necessity 
of avoiding certain critical numbers, and later of forcing the ex- 
perimenter to draw from these numbers regardless of what might 
be the initial number of beads presented for solution. 

Likewise in all series of a given order the lowest possible draw 
is a common element; e.g., in Series 1-2, 1-3, 1-4, etc. the low 



90 JOHN C. PETERSON 

draw is the same for all series. It is clear that this common 
element is responsible for a large portion of the transfer, for 
after the solution of Series 1-2, in which neither the high nor 
the low draw was usually recognized as especially related to the 
formula, it required only 41.7 trials on the average to generalize 
for all possible variations of the high draw with the low draw 
remaining constant. But it required 130.3 additional trials on 
the average to generalize for all possible variations of both the 
high and the low draw.^^ In other words, the mastery of an 
indeterminate number of series in which one of the two most 
essential elements is identical, required less than one-third as 
many trials as were necessary for the mastery of an indetermi- 
nate number of series in which this objectively identical element 
is lacking. And this notwithstanding a large amount of transfer 
through other elements from the former to the latter group of 
series. Other common elements of this sort might be mentioned, 
as for example, the different sequences of E's draws (described 
above, p. 18), but with further enumeration it becomes increas- 
ingly more difficult to distinguish between subjective and objec- 
tive identities. 

It must not be supposed that these objectively identical ele- 
ments were invariably recog"nized at once as common to the 
various problems of the series. Even the most obvious of them 
were often very slow in impressing themselves upon the minds 
of some subjects. But the absence of explicit awareness of these 
elements is no warrant for the contention that no transfer was 
mediated by them in the early stages of the experiment. The 
influence of unrecognized factors of some sort is clearly evi- 
denced by the fact that subjects were frequently able to solve 
problems though unable to say how it was done or to give better 
reasons for correct draws than that they "seemed right." 

b. Subjectively Identical Elements.. — Many of the more im- 
portant elements which came to be regarded as common to a 
number of problems are not, however, present in the same ob- 
jective sense as those mentioned above. Such identical ele- 

"9 The intermediate draws may be neglected since they varied in the same 
manner in both cases and generally attracted little or no attention. 



HIGHER MENTAL PROCESSES IN LEARNING 91 

ments are, in large measure, the result of the generalizing activi- 
ties of the subject. Judd has insisted at length upon the impor- 
tance of this type of common elem.ent as a source of transfer.^" 
Elements of this sort are fairly numerous and constitute the prin- 
cipal medium of transfer as found in this study. The high draw, 
or H, for example, is clearly dependent upon generalization for 
its status as a common element in all series, as is also the low 
draw, or L. These two symbols with their associated meanings 
are undoubtedly the most effective instruments of transfer for 
all subjects who mastered any considerable number of series. It 
is through the medium of these two minor concepts, largely, that 
the meanings developed in and about specific critical numbers, 
become detached and generalized and finally applied to other ap- 
propriate elements in the same and other series. On the basis of 
these and other generalized elements every subject, who had the 
perseverance to stay with the task, sooner or later accjuired the 
ability to solve at sight series in which neither the low nor the 
high draw was common to the preceding series in a purely ob- 
jective sense. 

Common elements of these two types were not always easy to 
distinguish, however. Specific elements of the objective sort 
were often apprehended in general terms at their first presenta- 
tion, or, at any rate, the subjects' verbal reactions to them were 
couched in general terms. Thus the fact that, after reducing the 
number of beads to a multiple of L + H, E always drew i when 
S drew 2 and 2 when S drew i, was often noticed and alluded 
to by the subject as "drawing opposites." 

c. Generalized Methods of Procedure. — Closely related to the 
generalized elements of content and depending largely tipon them 
were certain generalized methods of procedure. The most com- 
mon of these types of procedure to develop in the first series and 
persist through later series, consisted of some sort of systematic 
alternation of various sequences of draws designed to prevent 
useless repetition of unsuccessful variations or to exhaust the 
possibilities of variation. One subject described his procedure 
as follows : "First I take i and then the same as you took ( for 

4° Judd, Psychology of High School Subjects, p. 414. 



92 JOHN C. PETERSON 

the remaining draws of the trial), then i and the opposite; then 
I take 2 and the same, and finally 2 and the opposite." The first 
and third of these sequences of draws were not successful. The 
others were successful approximately half of the time upon non- 
critical numbers. 

There is good evidence that the erroneous sequences were 
brought in by a mere contrast association between the words 
opposite and same. Here are the circumstances : After winning 
twice in succession from 14 beads, the subject said: "If I take 
I when you take 2 and 2 when you take i, I win; i.e., if I take 
the opposite of your draw." After three trials upon 15 beads 
he said : "The principle I discovered seems to hold now for 
you." He then began immediately to follow the experimenter's 
draws, i.e., to draw the samie. This failing in the first three 
trials, he changed again to the opposite in the fourth trial, and 
added at its completion: "When that was reduced to 12, I tried 
taking the opposite of what you took, but it didn't work." He 
then began to follow the procedure described above and contin- 
ued for over an hour to the very obvious detriment of his prog- 
ress. Finally the erroneous sequences were eliminated and hia 
progress somewhat accelerated. This modified procedure was 
carried forward to later series. 

Another helpful procedure which was used quite effectively by 
some of the subjects was to begin the search for critical numbers 
with o which was regarded as the first critical number in every 
series. This had the effect of limiting the number of problems 
to be solved in locating a sufficient number of critical points for 
generalization. The procedure was adopted by Subject vi at the 
beginning of Series i or 6 and by Subject ii near the beginning 
of the experiment. 

An effective method of limiting the scope of attention in the 
discontinuous series without sacrificing any useful variations, 
was to subtract the highest multiple of L + H contained in the 
number presented for solution, and work upon the remaindei 
which was thus brought well within the range of perceptual con- 
trol. In Series 2 or 5, for example, if 17 beads were presented, 
the subject would find the difference between 14 and 17 and deal 



HIGHER MENTAL PROCESSES IN LEARNING 93 

directly with this small remainder. After the first draw had 
been determined in this manner it was usually possible to make 
the remaining draws mechanically upon the principle of drawing 
opposites. This procedure was followed more or less persistently 
with good effect by about a third of the subjects. 

A less definite though perhaps more profitable form of syste- 
matization consisted in a better balancing and correlation of ra- 
tional thinking with overt trial-and-error procedure as the ex- 
periment progressed. In the early problems much of the manipu- 
lation was extremely ill-directed, or undirected, and much time 
was wasted in fruitless attempts to generalize upon insufficient 
data. The later vv^ork of most subjects showed much improve- 
ment both in the purposeful direction of trial-and-error pro- 
cedure and in the matter of judgment as to when enough data 
were in to warrant generalization. This process of empirically 
working out a proper balance and correlation between reflective 
thinking and overt trial and error, is probably one of the main 
sources of progress in all successful attempts to deal at all ex- 
tensively with novel data. 

d. Effect of Thoroughness of Learning upon Transfer. — It 
might be expected that the superior ability which enabled one 
subject to outrank another in the mastery of Series 1-2 would 
give him a proportionate advantage in the solution of the later 
series. As a matter of fact, however, subjects whose speed of 
progress in the solution of Series 1-2 is above the median show 
no superiority in the immediately following series over those 
whose speed is below median in the first series. This is true of 
all groups of subjects who solved as much as two- successive 
series of problems, as is shown in the following data. The rate 
of progress is expressed in terms of the number of trials re- 
quired for the solution of a series : 

Although the subjects whose rate of progress was below me- 
dian required over five times as many trials for the solution of 
the first series as was required by those whose rate was above 
median, they were able to solve the second series in practically 
the same number of trials as was required by their presumably 
more gifted fellow subjects. The degree of transfer from the 



94 



JOHN C. PETERSON 



Group I 
(14 subjects) 
First Second 
series series 



Group IIA41 

(7 subjects) 

First Second 

series series 



Group IIB 

(10 subjects) 

First Second 

series series 



Total trials of sub- 
jects above median*^ 
Total trials of sub- 
jects below median 



2333 



144 



182 



307 



1638 



318 



64 



75 



83 



First 


Second 


series 


series 


1113 


311 


5849 


329 



Total trials required by 16 subjects above median, 
Total trials required by 16 subjects below median, 
*i The record of Subject J is here omitted because of an irregularity of 
procedure mentioned above on page no. 

42 Above median here means above median speed of progress as in the 
foregoing text. 

first to the second series, in terms of the number o£ trials re- 
quired, was on the average 94 per cent for the subjects whose 
rate of progress in the first series was below median and only 
72 per cent for those whose rate was above median. That these 
group averages do not grossly misrepresent the individual rec- 
ords is evident from the fact that only 4 of the 16 subjects whose 
rate of progress was below median in their respective groups in 
the first series, were surpassed in the percentage of transfer by 
any of the 16 subjects whose rate was above median. Correla- 
tions by the rank method between the quickness of mastery of the 
first series, as measured by the number of trials required, and the 
degree of transfer from the first to the second series, give coeffi- 
cients of — .803, — .914, and — .819 for Groups I, IIA, and 
IIB respectively. 

The thorough acquaintance with the elements of the problems, 
which is acquired by the initially slow subjects during their long- 
continued efforts to find a solution, is undoubtedly the main cause 
of the relatively high degree of transfer found at this stage of 
the work of these subjects. The records show clearly that the 
initially slow subjects generally surpassed the more speedy ones 
in the number and variety of elements which were discovered in 
the first series. From their four-fold more numerous responses 
to the same objective situations, it is not unreasonable to suppose 
that many of the associations formed by the slower subjects 



HIGHER MENTAL PROCESSES IN LEARNING 95 

were more firmly fixed than corresponding associations formed 
by the subjects whose progress was more rapid. The difference 
between the subjects whose initial progress was slow and those 
who progressed rapidly at the start, is not so much a matter of 
difference in the speed of formation of new associations as in 
the power to utilize these associations for a definite purpose. 
But the ready utilization of associations in the early stages of 
their development results in the accompli shmen'jt of the end 
before a sufficient number of repetitions has been made to fix 
the associations thoroughly. It also limits the number and va- 
riety of associations formed. Therefore the points of contact 
are usually neither so intimate nor so numerous, and transfer at 
this point is generally not so great, when the initial progress is 
rapid as when it is slow. 

3. Negative Transfer 

Numerous facilitating and inhibiting factors enter in at every 
stage of the work to determine the rate of progress of subjects. 
On the whole, the facilitating influences so far outweigh the in- 
hibiting ones as generally to make it difficult to detect the pres- 
ence of the latter, to say nothing of the accurate measurement 
of their effects. Occasionally, however, the presence of these 
negative factors becomes clearly evident from the comments of 
subjects, and sometimes their effects become so^ obvious as to 
make a rough quantitative statement of their strength possible. 
Thus it has been shown above (pp. 80-81) that the progress of 
Subject ii through Series i or 4 and i or 6 was only approxi- 
mately one-third as rapid as might reasonably have been expected 
but for the interference of certain superficial associations which 
were formed in the process of solving a few of the preceding 
series. Two clear cases of negative transfer were also given 
above (pp. 79-80) from the record of Subject viii. 

A very similar case of negative transfer from Series 1-2 to 
Series 1-3 is found in the record of Subject ii. In his generaliza- 
tion upon the data of the former series he said : "I can't win 
any multiple of 3." When presented with the first problem on 
Series 1-3 he said at once: 'T suspect right away that it will be 



96 JOHN C. PETERSON 

a multiple of 5 here — 3 plus 2 equals 5," After his first trial, 
however, he said: "I was mistaken. I see now that 3 plus 2 
plus I equals 6. Evidently I can win multiples of 5. It must be 
multiples of 6." This idea was wrongly confirmed by his first 
trial upon 6 which was a failure. His pleasure at the seeming 
confirmation of his theory was evident from the tone of his re- 
marks. "By Jove," he said, "you won that!" The persistence 
of this hypothesis is shown in his later attempt to account for 
his inability to win 8 beads. "If I take i," he said, "you might 
take I and leave 6, and I can't win 6, so I can't win 8." The per- 
sistence of this notion is again apparent in his final attempt to 
generalize at the conclusion of the series : "I add the numbers 
which one is permitted to take and find that the multiples of the 
resulting sum cannot be won." This significant slip of the 
tongue was quickly corrected, however, as follows: "No! No! 
I add the last number to the first because that would give you 
the total number which could be taken with two draws." 

If the first two series are equal in difficulty, the algebraic sum 
of all positive and negative transfer effects was only 31.4 per 
cent for this subject, though the average for all other subjects 
amounted to 89.1 per cent, and for the other six above-median 
subjects of Group I, 73.3 per cent. If the ratio of the number 
of draws required for the solution of the second series to those 
required for the first series had been the same as for all other 
subjects combined, he would have solved Series 1-2 in 4 trials. 
If this ratio had been the same for him as for the other six 
above-median subjects of Group I, he would have solved the 
series in 10 trials. Thus, figuring on a conservative basis, the 
negative transfer occasioned by the false analogy between the 
two series, and perhaps by some other inhibiting factors, was 
equal to 13 trials, or more than one-third the number required 
for the solution of the first series, and more than half the num- 
ber required for the second series. 

At the conclusion of Series 1-3 Subject iii generalized for all 
series as follows: "In general, you have got to add i to the 
highest number you may draw, and multiples of that sum cannot 
be won, e.g., for Series 1-3 it would be multiples of 4, for Serieis 



HIGHER MENTAL PROCESSES IN LEARNING 97 

1-4 it would be 5." Later in generalizing for Series 2-3 he 
said: "Combinations of 5 and i above are losers, i.e., numbers 
ending in i or 6." In Series 2-4 this subject did very well at 
the beginning. At 10 beads he said: "I win by leaving 6." 
Later when working with 1 1 beads he reiterated in more general 
terms : "When the starter can leave 6 beads, he wins it." After 
two trials upon 12 beads he added: "Combinations of 6, 12, etc., 
if they can be left by the one who draws last, win ; i.e., the player 
who can leave 6, 12, etc. wins. After the second trial upon 13 
beads the subject generalized as follows : "Combinations of i 
above any nmiibcr'^^ or i above any combination lose." He was 
asked to give an example and replied : "Well, 4 plus i equals 5. 
Combinations of 5, 10, 15, etc.; or combinations of 6, as, 6, 
12, — no, combinations of 5 and i above, as, 6, 11, 16, etc. . . . 
anything that you can take away enough from to leave 5 you 
win." 

Note how neatly the idea, "Add i to the highest draw," slips 
in from the generalization for all series of the first order to 
vitiate his reasoning while he is attempting to apply the "one- 
above" element from the formula for Series 2-3. Such a con- 
fusion could never have occurred if in the first formula "Add i 
to the highest draw" had been generalized into the more widely 
applicable form, "Add L to the highest draw" ; that is to say, if 
the I to be added to the highest draw had been properly asso- 
ciated with the lowest possible draw in each series. Such asso- 
ciation would certainly have prevented the addition of i to 4 in 
Series 2-4 to find the number whose multiples are critical num- 
bers. The erroneous application of this idea is perhaps due in 
part to the similarity of its wording to that of the "one-above" 
element of the formulation for Series 2-3, for it was during the 
attempt to apply the latter element to Series 2-4 that the former 
one insinuated itself into the process. In the last instance, it 
will be observed, the wording of the two ideas is exactly the 
same, i.e., "i above any number" and "j above any combination." 

*3 Probably meaning i above the high draw in any series, since he had 
already generalized the high draw but had said nothing concerning the low 
draw. 



98 JOHN C. PETERSON 

The subject's illustration shows clearly that this ambiguous 
wording referred to the two diverse elements mentioned above. 
A somewhat similar case of negative transfer occurs in the 
transition from Series 3-4 to Series 3-5 in the record of Sub- 
ject xiii. The reader will recall that the groups of primary 
critical numbers are always composed of L critical numbers each, 
and that the interval between such groups of critical numbers is 
always equal to H. When, therefore, a new series is formed by 
adding i to the high draw of the old series, the interval between 
successive groups of primary critical numbers is increased by i. 
The groups of critical numbers, however, remain unchanged in 
size. But if a new series is formed by adding i to the low draw 
of the old series, the magnitude of the groups of critical numbers 
is increased by i and the interval between successive groups re- 
mains unchanged. The subject's error at this point consisted in 
applying the latter principle instead of the former. At the con- 
clusion of the preceding series he had said : "The critical num- 
bers are multiples of 7 and i and 2 beyond these multiples." At 
the presentation of Series 3-5 he said immediately: "This will 
be multiples of 7 and i, 2 and 3 beyond." This attempt to apply 
the wrong principle is the only plausible explanation of the 
negative transfer which occurred at this point. There were no 
interruptions and no apparent fatigue. Such incorrect choice 
among principles which are but vaguely understood and inse- 
curely held in mind recalls the difficulty encountered in the at- 
tempts to generalize upon the elements of a series in the early 
and unstable stages of their abstraction. These facts emphasize 
the importance of thoroughness in learning for positive transfer 
through generalization of experiences, and point to the danger 
of negative transfer from vagueness and under-learning. 

4. Summary 
The ability of subjects to solve the later problems of the ex- 
periment was much modified by the effects of learning in the 
mastery of earlier problems. Both facilitating and inhibiting 
factors were operative, though, on the whole, the former so far 
predominated as to leave a large balance of positive transfer. 



HIGHER MENTAL PROCESSES IN LEARNING 99 

This is true whether successive problems in a series, successive 
series of the same order, or successively higher orders of series 
are taken as a basis of comparison. 

So far as our analysis goes, the greater the objective similarity 
between successive problems or series of problems, the higher is 
the degree of transfer. But it is not always possible to distin- 
guish between objectively identical elements and common ele- 
ments which owe their identity to the generalizing activities of 
the subject in apprehending them. It is, however, clear that a 
very large portion of the transfer observed, was effected through 
the medium of common elements of the latter sort, i.e., through 
concepts, and general principles, methods, attitudes, etc. 

Some difficulty was encountered in the later series in attempts 
to apply generalizations from preceding ones. Misapplication 
of such generalizations often resulted in considerable loss of time 
and effort. False analogies, which were found to be the prin- 
cipal source of erroneous applications, resulted in the main from 
incompleteness of analysis either of the old problems upon which 
the generalizations were based, or of the new problems to which 
they were to be applied. Another source of negative transfer 
was found in the occasional ambiguous wording of generaliza- 
tions, resulting in either their distortion or misapprehension. 

There was, on the whole, much more transfer from the first 
to the second series in the work of subjects whose initial speed 
of progress was slow than in that of subjects who were more 
speedy at the start. This, apparently, was due in large measure 
to differences in the thoroughness of the acquaintance acquired 
with the elements of the first series. 

G. Effect of Age and Education 
Though the data at hand do not warrant a separate discussion 
of the effects of age and education on each of the processes of 
abstraction, generalization, and transfer, it may be worth while 
briefly to present such data as we have regarding the influence of 
these factors upon the general ability to solve our problem. Only 
two of our subjects differed enough from the general level of 
the group to warrant any attempt at comparison upon this point. 



100 JOHN C. PETERSON 

One of these subjects was in the preHminary group. The other 
is Subject iii of our major group of subjects." The record of 
the latter will be mentioned first. 

Subject iii of Group I was a thirteen-year-old boy who was in 
about the middle of his first year in high school. 'T believe he 
is the brightest boy I have ever known," was the reply of his 
teacher in mathematics when asked concerning the mental ability 
of the boy. The father of this boy has shown remarkable genius 
in the business world. A comparison of the record of this sub- 
ject with those of nine other subjects of Group I can be made by 
turning back to Table III. It will be noticed that Subject iii 
solved all the problems of the entire experiment in 8,425 seconds 
— considerably less time than was required by any other subject. 
Subject i, the next speediest in point of time, required 11,215 
seconds, or 34.3 per cent more time than was required by the 
younger competitor. However, Subject i required only 165 
trials whereas Subject iii required 388. Subject ii also finished 
with fewer trials than were required by Subject iii. Again, 
fewer individual problems were solved by each of these adult 
subjects than by Subject iii before a satisfactory generalization 
for all series could be formulated. Thus the lead in time which 
Subject iii gained over Subjects i and ii was lost in the greater 
number of reactions required for the attainment of a given de- 
gree of mastery of the problems. Subject iii appeared to be less 
conscious of method, somewhat more reliant upon trial-and-error 
procedure, and quicker in reaction than Subjects i and ii ; but in 
no respect did his responses vary so much from those of these 
two adults as did those of some of the other adult subjects. 

The other boy who participated in the experiment was eleven 
years old and had just completed the work of the seventh grade 
in the elementary schools. This subject served in the preliminary 
experiments in which fewer of the discontinuous series were 
given. His record upon these series is not, therefore, strictly 
comparable with those of any of the later subjects. Fortunately 

^*With the exception of one senior college student all of the adult sub- 
jects of Group I were graduate students or instructors at the University 
of Chicago. 



HIGHER MENTAL PROCESSES IN LEARNING loi 

the father of this boy, a doctor of philosophy, solved the same 
problems as the son in the same order. . Hereafter we shall refer 
to the son as K and to the father as J. The only difference in the 
procedure with these two subjects was that J did the work in 
two sittings on successive days whereas K worked five successive 
days but with much shorter periods. It does not seem probable, 
however, that the relative speed of progress of the two subjects 
was seriously affected by this difference in the length of periods 
of work, since J stopped work in each instance as soon as he 
began to feel fatigued. The records of these two subjects are 
given in Table XVI. 

Table XVI 
Summary of Records of Subjects K and J. 











Subject K 




Subj 


ect J 












No 


Problems 






No. 


Problems 


Series 




No. 


Trials 


Solved 


No. 


Trials 




Solved 


1-2 






34 




9 




34 




17 


1-3 






13 




6 




18 




6 


1-4 






I 




I 


. 










1-5 






o 

















2-3 






4 




2 




17 




10 


2-4 






3 




2 




45 




— . 


2-5 






8 




2 




4 




2 


3-4 






2 




2 




10 




8 


3-5 






3 




I 












3-6 






o 

















5-6 






I 




I 












3 or 


I 




6 




4 




5 




5 


2 or 


6 




22 




16 




8 




8 


3 or 


9 




13 




9 




6 




9 


2 or 


7 




36 




21 




14 




16 



Total 146 76 116 81 

*5 Accidentally omitted. 

There is no marked difference in the work of these two sub- 
jects. They required the same number of trials for the first 
series, after which K took the lead throughout all of the remain- 
ing continuous series but was surpassed by J in the discontinuous 
series. However, the lead established here by J was so small 
that it might easily have been lost if the work of the two sub- 
jects had continued further along parallel lines. 



102 JOHN C. PETERSON 

It is worthy of note that this eleven-year-old boy solved all of 
the continuous series in 69 trials though the best record estab- 
lished by any member of group I, that of Subject ii, was 79 
trials. The differences in procedure in the work of K and that 
of Subject ii were in the length of work periods and in the fact 
that K (as also all other subjects who served in the preliminary 
experiments) did not have the stop-watch, dictaphone, and met- 
ronome before him and he worked with matches instead of beads. 
The effect of these differences upon the rate of progress was 
probably not great. 

Considering the complexity of the mental processes involved 
and the limited amount of data at hand, it would be rash to at- 
tempt an explanation of the essential equality in the achieve- 
ments of these children with those of the most capable members 
of our highly selected group of educated adult subjects. Fur- 
ther experiments are now being undertaken upon comparable 
groups of children and adults with a view to the analysis and 
comparison of their reactions to various sorts of novel problems. 



IV. DISCUSSION AND CONCLUSIONS 

For convenience of treatment we have divided the learning 
process here under study into three stages : ( i ) the abstraction 
of elements from the problematic situation, (2) the combination 
of essential elements into higher conceptual units, and (3) the 
application of these higher unitary processes to situations other 
than those out of which they arose. Some elements advance 
into the later stages before others emerge into the first, and 
some telescoping of successive stages occurs with many of the 
elements, so that there is considerable overlapping of stages in 
the process of solution as a whole. Yet these stages are char- 
acteristic enough of the growing adjustment to individual ele- 
ments and to the problem as a whole to warrant their separate 
treatment. 

Each stage begins with diffusion and multiple response, and 
progresses through elimination and the preservation and gradual 
co-ordination of essential responses. Trial and error is the 
method of procedure in all stages, but the materials among which 
variations occur are somewhat different and the field of varia- 
tion is progressively narrowed in the later stages. 

A. Abstraction of Elements. — Probably none of the elements 
of the problems were wholly new to any of the subjects but the 
essential elements were sufficiently mingled and fused with un- 
essential ones to require considerable analytic activity before the 
solution of a series was possible. The prevalence of random 
manipulation in the first stage and the notable failure of all at- 
tempts at anticipatory analysis, are in accord with Ruger's ob- 
servations upon the close external similiarities existing between 
human and animal methods of learning.^ Our data enable us, 
in a modest measure, to carry the comparison beyond the super- 
ficial resemblances into the selective factors which determine the 
course of learning. This is not an easy task, however, owing to 
the diversity of opinion among experimental psychologists con- 
1 Op. cit., p. 12. 



104 JOHN C. PETERSON 

cerning the selective factors in animal learning, and to the nec- 
essarily tentative nature of inferences as to the factors involved 
in so limited a study as our own.^ 

Perhaps the most important factor in the abstraction and se- 
lection of the essential elements of our problems, is frequency of 
repetition of responses to those elements. It has been shown 
(see above, pp. 40-47) that attention quickly came to be largely 
dominated by those elements of the problematic situation which 
were most often repeated. Likewise those elements which served 
most frequently as stimuli were the first to acquire definite mean- 
ings and special value for purposes of control. 

Usually the most frequent objective element of response was 
the subject's notation of the number of beads remaining before 
each of his own draws. Though this type of counting did not 
occur with all subjects — at times some counted the numbers al- 
ready drawn out by either the subject or the experimenter or by 
both, or seemingly did not count at all — yet it appeared to be the 
dominant form of response of the rapid learners, and indeed of 
all learners during periods of rapid progress. This counting re- 
action to the various numbers presented was clearly a basic ele- 
ment in the formation of the associations upon which the critical- 
number meanings were founded. It cannot be maintained that 
the frequent repetition of this important element of response was 
merely an effect rather than one of the causes of learning, since 
the frequency of its occurrence tended rather to be diminished 
than to be increased by learning.^ 

Some doubt has been raised as to the possibility of change in 

2 See Watson, "Behavior," pp. 267-268, and Joseph Peterson, "The Effect 
of Length of Blind Alleys on Maze Learning," Behav. Mon., 1917, 3, No. 4. 

2 Speaking of the status of frequency and recency as factors in maze 
learning Joseph Peterson says : "It cannot be too strongly pointed out . . . 
that this increasing percentage of reactions agreeing with the expectations 
based on recency and frequency effects, as learning advances from the first 
random stages toward the establishment of a regular habit, cannot be safely 
regarded as evidence that learning is brought about by recency and fre- 
quency factors : our evidence seems to justify the contrary conclusion, that 
this increase in reactions favoring recency and frequency factors is the 
result of the learning." (Peterson, Joseph. "Frequency and Recency Fac- 
tors in Maze Learning by White Rats," p. 359.) 



HIGHER MENTAL PROCESSES IN LEARNING 105 

the order of responses in maze learning through the operation of 
frequency factors.* Even if the argument is well-founded, it is 
difficult to see how it could affect the explanation of learning in 
the present study, since both the direction of attention towards 
the critical numbers and the growth of the critical-number mean- 
ings may well be regarded as the effect of the mere formation and 
strengthening of associations between simultaneous or successive 
elements of stimulation and response. There is no necessity for 
change in the order of functioning of these associations. 

It need hardly be said that we do not mean to contend that 
analysis proceeds upon the basis of the principle of frequency 
alone. As previously pointed out, the order in which the various 
critical numbers were abstracted is almost the exact order of 
their nearness to the goal, or end of the trial. This is in accord 
with the observations upon the maze-learning of rats. Com- 
menting upon Hubbert's experiments, upon the basis of which 
backward elimination had been denied, Carr says : 

Determining the average number of trials necessary to elimi- 
nate each cul de sac, the order of elimination for the entire group 
was 6-5-3-4-2-1, where the successive errors are numbered in 
order from the entrance. This order gives by the rank method 
a positive correlation of .943 between quickness of elimination 
and propinquity to the food box. The average number of trials 
necessary to eliminate the last three errors was less than that for 
the first three for 90 per cent of the rats. Surely there is a very 
pronounced tendency for the errors to be mastered in proportion 
to their nearness to the food box, and the deviation from the 
exact correlation for each rat may well be due to the operation 
of other causal agencies.^ 

In summarizing her data concerning the order of elimination of 
errors, Vincent says : 

The elimination of the final members of the series first is not 
only true of the groups as a whole but also of the individual 
animals.*' 

^Ibid., pp. 347-348. 

^ Carr, H. A. "The Distribution and Elimination of Errors in the Maze," 
Jour. Animal Behav., 1917, vol. 7, p. 146. 

6 Vincent, Stella B. "The White Rat and the Maze Problem," Ihid., 1915, 
vol. S, p. 371. Regarding the applicability of Vincent's data Hubbert and 
Lashley say : "In the form in which this is presented, however, Vincent is 



io6 JOHN C. PETERSON 

Peterson finds the same general backward elimination of errors/ 
and even Hubbert and Lashley agree that 

when the averages of very large groups of animals are taken, 
there does seem to be a progressive elimination of errors from 
the food compartment to the entrance of the maze.* 

What are the factors upon which the regressive order of mas- 
tery of the elements of the maze and of our own problems, is 
based? Though some of the suggested agencies in the determi- 
nation of the order of learning in the maze are obviously inap- 
plicable to our problems, it is not improbable that the determin- 
ing factors are in part identical in the two forms of learning. 
It may be said at once that we do not assume with Watson, that 
the demonstration of a backward order of mastery need give any 
comfort to the advocates of the retroactive effects of pleasure 
as a causal factor in learning. As a matter of fact, the goal of 
most trials of our subjects was defeat; and here, as elsewhere, 
defeat usually gave rise to considerable evidence of disappoint- 
ment or annoyance. 

It may be argued that no explanation is necessary, beyond the 
greater frequency of response to lower than to higher critical 
numbers. This argument would be in essential agreement with 
Peterson's explanation of the order of mastery of the elements 
of the maze.'' That other factors entered into the determination 
of the order, however, is clearly evident from the records of 
some of the subjects of Group III. While wrestling with 14 
beads as their initial problem in Series 1-2, some of these sub- 
jects made nearly all of their errors in drawing from 14, or 11 
beads; and, in consequence, reacted practically as often to 9 as 
to 6 beads. Yet 6 was recognized as a critical number some time 
before 9 was so regarded. Further evidence that the order of 

not justified in applying the data to the problem. She makes no statement 
as to what constituted a trial in her experiments and in two 'typical records' 
. . . we find that the rat, after reaching the food, was allowed to return 
and re-explore the maze." (Ibid., 1917, vol. 7, p. 132.) 

^ Op. cit., pp. 360-361. 

® Hubbert and Lashley, "The Elimination of Errors in the Maze," Jour. 
Animal Behav., 1917, vol. 7, No. 2. 

9 Peterson, Joseph, op. cit., pp. 338-346. 



HIGHER MENTAL PROCESSES IN LEARNING 107 

learning was affected by the relative nearness of the elements to 
the goal, is found in the records of two subjects for whom the 
procedure was so modified as to eliminate the frequency factor, 
except in so far as this factor is itself determined by proximity 
to the goal/** All of the critical numbers learned by these two 
subjects were learned in the exact order of their nearness to the 
goal. 

Attempts to solve the higher numbers by merely hitting upon 
a correct sequence of draws by a sort of long-range trial and 
error, had played a minor role in learning in the former experi- 
ments, but such attempts were practically useless here. Progress 
in analysis, which had previously been confined to the elements 
near the goal, was here almost entirely so confined. 

At the beginning of their work our subjects (and this is true 
of all the groups) were able to foresee the outcome of a trial 
from a distance of not more than the first or, rarely, the second 
critical number above the goal, and that only with some doubt 
and uncertainty. But with continued repetitions the foresight of 
consequences became clearer and more certain until finally the 
critical number next above the goal became so closely associated 

10 The first of these subjects, a boy of fourteen years of age, worked upon 
30 beads in Series 1-3 as his initial problem. To avoid the suggestive se- 
quence of critical numbers in exact multiples of 4, success in a trial was 
defined as forcing one's opponent to draw last. The critical numbers then 
become i, 5, 9, 13, etc. Again, the simple, suggestiive sequences of the ex- 
perimenter's draws, followed in the earlier experiments, were abandoned in 
favor of more difficult sequences which would tend to resist formulation and 
so impel the subject to attend more exclusively to the number remaining 
before each draw. 

This problem was found to be too difficult. After 7;^ trials requiring up- 
wards of two hours of time the subject had discovered only 5, p, and 13 of 
the critical numbers above i, and had made practically no progress in the 
organization of the few facts observed. 

A simpler problem, consisting of a similar variation of Series 1-2 with 25 
as the initial number, was therefore given to the second subject. This sub- 
ject succeeded, after 72 trials occupying a little less than two hours, in dis- 
covering all the critical numbers in his problem and in formulating his dis- 
coveries into a suitable generalization for the entire series. The number of 
draws from the various critical numbers was here almost exactly uniform, 
and all of the critical numbers were discovered in the exact order of their 
magnitude. 



io8 JOHN C. PETERSON 

with defeat as to constitute a new point of orientation from 
which to push the analysis back one step farther. Thus, after 
its discovery and a sufficient number of repetitions to associate 
it firmly with the next lower critical number, each critical num- 
ber became in effect a new goal. Two important consequences 
follow from this fact: (a) the next higher critical number was 
brought within range of direct and more or less clear apprehen- 
sion of the consequences of each of the possible reactions to it, 
and (b) the time intervening between the occurrence of each of 
the higher critical numbers and the full realization of defeat, was 
reduced by a number of seconds. 

It may be argued that (a) would result merely in the arousal 
of a new type of reaction to the critical number next above the 
new point of orientation, which, by frequent repetition, would 
gradually get itself established as the invariable response. This 
fact is freely admitted. But since the arousal of this sort of re- 
sponse is directly dependent upon the proximity of its stimulus 
to the goal, its repetition can hardly be used as evidence that 
frequency factors are alone responsible for the regressive order 
of mastery of the elements of the problem. On the contrary, the 
retroactive influence of the goal in selecting and consolidating 
the closely preceding essential reactions is here clearly evident. 

Whatever causal agencies may have been involved, there can 
be no doubt that the rate of formation of associations between 
the goal response (the positive recognition of defeat in the cur- 
rent trial) and the various critical numbers varied directly with 
the relative nearness of these numbers to the goal. There seems 
to be no good reason why association should not be facilitated by 
a close temporal approximation of the various elements to be 
associated. That such facilitation does occur is suggested by 
certain experimental observations by Carr and Froeberg upon 
animal and human learning respectively. Carr says : 

The final co-ordination consists of an association between each 
act and the sensory aspects of the preceding act as well as a dis- 
tinctive motor attitude resulting from the same. The relative 
efficiency of the two stimuli in determining the choice varies 
with the individual. The problem was mastered quickest by 



HIGHER MENTAL PROCESSES IN LEARNING 109 

those animals that relied mainly upon the factor of motor atti- 
tudes in making their choice. This fact suggests the hypothesis 
that the speed of learning is to some extent a function of the de- 
gree of temporal contiguity between the terms to be associated/^ 

Froeberg studied the effects of varying the length of the inter- 
val between the presentation of nonsense syllables in pairs, the 
interval being occupied by the repetition of two-digit numbers. 
Commenting on his results he says : 

There is thus a distinct, though irregular, decrease in the num- 
ber of right responses as the interval between the numbers of the 
pair in successive presentation increases. ^^ 

This view of the effect of the degree of temporal contiguity 
upon the speed of association is in accord with the close relation 
already observed between the speed of reaction of our subjects 
and the quickness of their discovery of the principle of drawing 
opposites. Moreover, the operation of such contiguity factors is 
evident in the fact that subjects commonly learned to recognize 
the critical status of numbers some time before they were able 
to give a reason for it. 

Thus in dealing with novel situations through the abstraction 
of essential situation- and response-elements and the organiza- 
tion of these elements into serviceable concepts, mere frequency 
of repetition appears to play a role of no less importance than in 
the selection and combination of essential elements of stimulus 
and response into effective sensori-motor co-ordinations. The 
mechanical operation of frequency and of the factors mentioned 
in (a) and (b) above appears also to be quite as effective in de- 
termining the order of mastery of the elements in this type of 
conceptual learning as it has been supposed to be in sensori-motor 
learning.^^ 

11 Carr, H. A. "The Alternation Problem," Jour. Animal Behav., 1917, 
vol. 7, No. 5, p. 384. 

12 Froeberg, Sven. "Simultaneous versus Successive Association," Psych. 
Rev., 1918, vol. 25, No. 2, p. 162. 

13 Under the conditions of maze learning the process mentioned in (a) 
need involve only direct responses of the conditioned-reflex type to present 
stimuli. This statement would appear to imply the operation of "retroactive 
association," the possibility of which has been questioned by Hubbert and 
Lashley {op. cit.) and by Peterson (Joseph Peterson, "Frequency and Re- 



no JOHN C. PETERSON 

B. Combination of Essential Elements. — In the early stages of 
the analysis of our problems, as at the beginning of the learning 
of telegraphy, typewriting, chess playing, etc., the individual ele- 
ments are apprehended in relative isolation. Attention to one ele- 
ment precludes attention to others at the moment, with the result 
that the responses are of a random, haphazard character. With 
the progress of learning the span of attention is gradually broad- 
ened by the combination and organization of individual elements 
into larger, more meaningful units through which it is possible 
to focus one's relevant experiences upon new situations with a 
minimum loss of time and effort. 

Regarding this process of combination and organization of 
essential elements Cleveland says : 

Sow organization can best be brought about is still an open 
question. ... Its ultimate nature we do not know. To a great 
extent the material organizes itself, i.e., the organization is physi- 
ological and a matter of growth.^* 

That this organization is very largely independent of explicit 
conscious direction is indicated by the fact that the critical status 
of numbers was usually recognized before the subject was able 
to give a reason for it; that the prinicple of drawing opposites 

cency Factors in Maze Learning by White Rats," Jour. Animal Behav., 1917, 
vol. 7, No. 5) though it is not entirely clear what connotation is to be given 
the term. If a "pleasurable situation" at the end of the series of responses 
to be learned is an essential factor, it is quite apparent that there was no 
retroactive association operative in the learning of our subjects. But the 
inclusion of this factor was seemingly not intended — though one cannot but 
suspect that the aversion to the idea of retroactive association arises from 
its early complication with the controverted stamping-in effects of pleasure — 
since, after its definition in terms of reactions leading up to a pleasurable 
situation, the process is restated in terms of conditioned reflexes. 

Hubbert and Lashley's evidence against the presence of retroactive as- 
sociation in maze learning is, to say the least, inconclusive. The effect of the 
marked progressive shortening of the true runways toward the center of 
the maze was not determined. Peterson (Joseph, op. cit., p. 361) has pointed 
out two other sources of weakness in their argument. Yet he writes (p. 
362) : "There seems to be no 'retroactive association' necessary, as Hub- 
bert and Lashley rightly conclude." His evidence for this more conservative 
view is, however, drawn from his own experiments. 

1* Cleveland, A. A. "The Psychology of Chess and of Learning to Play 
It," American Journal of Psychology, 1907, vol. XVIII, p. 297. 



HIGHER MENTAL PROCESSES IN LEARNING in 

was often followed and sometimes clearly formulated before 
the reason for it was known, and that solutions for series some- 
times obtruded themselves upon a subject's consciousness after 
a period of rest during which no thought had been given to the 
problem and prior to which the subject's grasp of the elements 
of the problem had been characterized by confusion and lack of 
organization. Moreover, the unreasoned recognition of the prin- 
ciple of drawing opposites, and the backward order of learning 
of critical numbers, are clearly explicable on the basis of fre- 
quency of repetition and close temporal contiguity of the ele- 
ments which were associated, as already shown. 

The essential elements were unified not so much by direct as- 
sociation from one element to another as by the association of 
each element with a common intervening symbol. The character 
and manner of selection of such a symbol is a matter of some 
importance. Sometimes, during the early stages of unification, 
the intervening symbol is merely one of the elements of the con- 
crete situation, or a direct response to such a situation-element, 
which happens to occur more frequently than others or to possess 
some other advantage, perhaps of intensity, duration, or posi- 
tion. Thus the goal of the trial, or the attitude aroused by de- 
feat, or the verbal expression of the latter appeared to serve as 
a common symbol in the early stages of analysis and organiza- 
tion. 

But as soon as enough numbers had been classified as "losing," 
"impossible," or "insoluble" to give the subject a general schem- 
atic impression of the character of the problem, the efforts at 
organization began to proceed more largely through the arousal 
and application of old concepts which functioned as a crude sort 
of hypotheses. Here the associations between the elements and 
the symbol were already in existence, provided the conception, 
or hypothesis, was correct. Henceforth the problem of unifica- 
tion became more a matter of calling up and testing out of vari- 
ous trial hypotheses than of the formation of new associations. 

The various concepts which, under the influence of the general 
conception of the problem, functioned as hypotheses, appear to 
have been aroused in three fairly distinct ways: (a) through 



112 JOHN C. PETERSON 

their association with a specific element of the problem, (b) 
through the combined associative pull of a number of discon- 
nected elements observed in rapid succession, and (c) through 
their association with relations observed to exist between indi- 
vidual elements, or, perhaps, with symbols representing such re- 
lations. 

As an Example of (a), a subject has conceived of 9 as an in- 
soluble number. It occurs to him that 9 is an odd number or 
that it is a multiple of 3, and he straightway infers that all odd 
numbers or all multiples of 3, as the case may be, are insoluble. 
This is the all but exclusive manner of origination of hypotheses 
in the early stages of learning when, owing to a lack of acquain- 
tance with the elements of the problem, the span of attention to 
the elements is very narrowly limited. Subjects who have passed 
beyond the early stages of learning also revert to this form under 
the stress of emotion, self-consciousness, or fatigue, which tend 
seriously to narrow the span of attention to the elements of the 
problem. There are some individuals who, through native limi- 
tations of the span of attention seem doomed to a marked de- 
pendence upon this primitive form of arousal of hypotheses. 
These subjects were prone to react hastily. They drew rapidly 
and, disregarding recently observed and often well-known facts, 
frequently jumped to unwarranted conclusions from which they 
showed little ability to extricate themselves by crucial tests. That 
the limitation of the span of attention exhibiting itself in this 
lack of inhibition is probably native is indicated by the fact that 
the reactions of these subjects continued to be relatively rapid 
and ineffective even after an acquaintance had been gained with 
a fairly large number of elements. There are, of course, some 
subjects who are able to deal both rapidly and effectively with a 
wide variety of elements. Subjects iii and xi were the most 
notable examples of this type. 

The origination of hypotheses in the manner described in (b) 
was observed in the later stages of learning, particularly in the 
work of subjects who inhibited the primitive tendency to general- 
ize upon every instance and were intent upon the discovery and 
retention of several critical numbers prior to generalization. 



HIGHER MENTAL PROCESSES IN LEARNING 113 

With repeated recall of such previously observed critical num- 
bers there was an increased facility manifesting itself in greater 
speed of repetition. Finally, after one or two repetitions of sev- 
eral critical numbers in rapid succession, the subject would sud- 
denly announce his belief that all multiples of 3 are insoluble. 
Likewise, the later discovery of the superficial relation of criti- 
cal numbers to the sum of the L- and H-draws often appeared 
to arise from the combined associative pull of a number of rap- 
idly reviewed instances the essential feature of which as sepa- 
rate instances were not clearly apprehended. Here also, as in 
dealing with the critical numbers of a single series, the first gen- 
eral formulation was usually given in a tentative manner and 
further evidence sought for its proof. But the further evidence 
by which proof was established was usually of exactly the same 
type as that which had preceded the formulation, namely, the 
enumeration of more confirmatory instances and the absence of 
refractory ones.^^ Sometimes hypotheses originated in explicit 
analysis and comparison of the elements of one or more series. 
This is the type of origin mentioned in (c) above. At its best 
this process involves a clear insight into the causal relations of 
the phenomena in question and carries its own confirmation. At 
its worst it results in false analogies of the sort mentioned in the 
discussion of negative transfer (see above, p. 95 ff.). This 
process is dependent upon a fair acquaintance with the elements 
of the problem and is therefore impossible in the early stages of 
adjustment to really novel situations. 

But the function of incipient hypotheses was not confined to 
the mere unification of previously discovered elements. These 
trial conceptions served also to direct attention to hitherto un- 
known elements and so aided materially in the discovery of new 
data." The latter function is particularly noticeable in case of 
such concepts as even and odd numbers, every other number, 
multiples of 3, etc. The verification of such an hypothesis 
through its appliation to previously known data and its function- 
ing in the discovery of new facts are but slightly different phases 

15 Cf. Dewey, "Studies in Logical Theory," p. 174. 

16 Cf. Dewey, op. cit., p. 145. 



114 JOHN C. PETERSON 

of the same process of application. Both can be explained upon 
the basis of the laws of habit/'^ For example, suppose that 15 
is under consideration and is conceived as a multiple of 3. 
Other multiples of 3 are immediately called to mind by associa- 
tion. But there is no invariable sequence. Sometimes, in ac- 
cordance with the habit of counting upwards, 18 and 21 are the 
first associates aroused ; sometimes the first associates to be 
aroused are 12 and 9, owing perhaps to the recency of their repe- 
tition. Simple associations of this sort probably account for 
much of the directive value of the more elaborate hypotheses of 
science. 

There is a notable difference in the explicitness and stability 
of hypotheses in different stages of their development. In- 
cipient hypotheses often occur in the early stages of the work as 
fleeting and perhaps vague insights. Further familiarization 
with the elements of the problem gives greater explicitness and 
stability to these hypotheses, though lapses still occur as is evi- 
denced by the subjects' failure to make even the simplest and 
most obvious applications. A degree of instability sometimes 
persists long into the later stages of the experiment owing to 
the rivalry of conflicting or superfluous hypotheses which, though 
entertained earlier in the game, may have been neglected in the 
meantime. Book found a similar instability in the building up 
of co-ordinations in typewriting. He says : 

It was observed by the learners that the older and more ele- 
mental habits used in the earlier stages of writing tended strongly 
to persist and force themselves upon the learner long after they 
had been superceded by higher-order habits. At every lapse in 
attention or relaxation of effort, the older habits stepped for- 
ward, as it were, and assumed control, thereby tending to per- 
petuate themselves. Only when a high degree of effort was be- 
ing permanently applied . . . was attention forced to lay hold 
of the higher and more economical methods of work." Refer- 
ring later to these fluctuations of attention and performance he 
says : "They were wholly beyond the learner's control. He 

1'^ This is in accord with Thorndike's view that "learning by inference is 
not opposed to, or independent of, the laws of habit, but is really their neces- 
sary result under the conditions imposed by man's nature and training." 
(Thorndike, E. L. "Educational Psychology," vol. II, p. 36.) 



HIGHER MENTAL PROCESSES IN LEARNING 115 

could not avoid them and could do little to regulate or control 
them."^' 

Continued attempts to deal with similar materials brought 
about a gradual strengthening of the associations necessary for 
the more ready arousal of appropriate concepts and the conse- 
quent failure of inappropriate ones to suggest themselves. This 
gradual automatization of the higher conceptual responses was 
observed by Cleveland in his study of chess. After giving an 
account of the combination of significant elements into larger 
complexes in which general terms "built up step by step" became 
increasingly more prominent, he says : 

We are in the habit of speaking of the automatic in the motor 
realm, meaning by it that certain movements or combinations of 
movements are carried on without conscious guidance. Is there 
such a thing as automatism in the realm of the purely intellect- 
ual? It seems to me that this question is to be answered in the 
affirmative. There is something in the purely intellectual life 
corresponding to motor automatism, which is shown in the abil- 
ity to think symbolically or abstractly, and thus to handle large 
masses of detail with a minimum of conscious effort. It involves 
the increasing ability to take in during a single pulse of atten- 
tion a larger and larger group of details which means, of course, 
that the attention is no longer needed for each one.^^ 

The combination of elements entering into a concept through 
various stages of automatization, is well illustrated in Fisher's 
study already referred to. Her subjects were presented with a 
series of ten somewhat similar figures exposed in succession for 
three seconds each. Each group was given a nonsense name, 
and the task consisted in defining the group name after the ob- 
servation of the series. When repeated exposures revealed no 
additional essential features the exposures were discontinued ; but 
the subjects continued at later sittings to recall the essential fea- 
tures of each group and report them as usual. Regarding the 
automatization of these concepts the author says : 

Functionally viewed (i.e., regarded from the point of view of 
the difficulty and effortfulness, or the ease and mechanizedness 

18 Book, "The Psychology o£ Skill," pp. 94, 122. 

19 Cleveland, "The Psychology of Chess and Learning to Play It," Ameri- 
can Journal of Psychology, vol. 18, p. 300. , 



ii6 JOHN C. PETERSON 

with which the concept-meanings entered consciousness), the re- 
calls ranged from an initial form in which more or less hesita- 
tion was present, to a final form which was marked by a high 
degree of mechanization and where the spoken statements fol- 
lowed in uneventful fashion, either immediately upon the in- 
structions themselves, or upon a brief and transitory visual or 
verbal image which served to 'set off' the train." Describing the 
final stage of the developments of the concepts under observation 
she continues : "At this stage the experience of generality was 
nothing more than an unhesitating, ready, and even mechanical 
mentioning of the general features. . . . The generality ex- 
perience was based essentially upon nothing more than a highly 
mechanized association between the words 'Zalofs are objects 
having,' — or 'Zalofs always have,' — and the enumeration of the 
essentials. The recalls were often given on a very automatic 
fashion." 

In recapitulation it may be said that, in the early stages of 
learning, those elements which occurred most frequently and in 
closest temporal contiguity were generally the first to be com- 
bined into higher units. Certain situation- and response-ele- 
ments which occurred most often and in closest proximity to 
other elements, became so closely associated with the latter as to 
serve as symbols through which the various elements were co- 
ordinated and their subsequent recall much facilitated. In the 
later stages of learning symbols representing well-organized con- 
cepts were called in by association. Through the medium of 
these symbols, and again by means of specific associations, mean- 
ings were transferred from previously known to newly dis- 
covered elements. Associations of this sort at first functioned 
slowly and imperfectly, but continued repetition brought about 
a facility and even automaticity of functioning comparable to 
that of sensori-motor co-ordinations. 

C. Application of Knowledge. — As previously mentioned no 
marked tendency was observed for perceptual solutions to find 
application beyond the limits of the specific concrete situations in 
which they occurred. This absence of transfer was attributed 
to the fact that responses were made directly to the objective 
situation rather than indirectly through the intervention of a 
symbol. The solution of numbers beyond the range of direct 



HIGHER MENTAL PROCESSES IN LEARNING 117 

perceptual control showed, on the other hand, a very marked 
tendency to apply to new situations. This fact was attributed to 
the intervention of symbols, which was necessitated by the extent 
and complexity of the materials to be controlled in the attainment" 
of a solution. 

The reason for the superior applicability of solutions involv- 
ing the use of symbols is not far to seek. The numerical sym- 
bols used in counting the beads remaining before each draw have 
been more frequently repeated than the concrete situations un- 
derlying the perceptual solutions and are therefore more easily 
recalled. Because of the greater ease of recall of often-repeated 
symbols than of seldom-observed situations, whatever meanings 
have been detached from the latter and associated with such sym- 
bols are more easily aroused in the presence of new situations. 
This advantage in the facility of recall of general symbols is 
important in as much as the conditions of application generally 
require the recall of the materials to be applied, while attention 
is directed primarily to the present problematic situation. 

But it is not through the greater ease of recall of meanings 
which have grown out of the problem situation and become at- 
tached to the symbols so much as through the arousal of pre- 
viously formed associations that such symbols serve to facilitate 
the application of old experiences to new situations. The sym- 
bols here employed in the reaction to all problems above 6 or 7 
beads, were necessarily numerical symbols which were already 
organized into various orderly sequences. The moment such a 
symbol is used each of these sequences tends to be aroused 
through association, and thus to extend the meaning of the term 
under momentary consideration to the other terms of the sug- 
gested series. Thus 12, pronounced insoluble and conceived as 
an even number, tended strongly to suggest the insolubility of 
other even numbers such as 10 and 14. But, conceived as a 
multiple of 3, it tended to suggest the insolubility of 9, 15, etc. 
The tendency for meanings to apply themselves to new data 
through the medium of such general symbols was often so 
strong as to distort the memory of frequently observed facts.* 
Thus one of the preliminary subjects, observing the insolubility 



ii8 JOHN C. PETERSON 

of 1 8 beads in Series 1-2 and conceiving it as an even number, 
immediately lost his somewhat feeble grasp upon previously ob- 
served facts and "remembered" positively that 16, 14, 12, and 
10 were also insoluble numbers. 

The range of application is obviously determined largely by 
the generality of the symbol through which transfer^" is to be 
effected, i.e., by the number and variety of particulars represented 
by the symbol. When, for example, the critical-number mean- 
ings were associated with 6, they were applicable to but one 
number in a single series. When these meanings became asso- 
ciated with the more general symbol, a-multiple-ofs, they be- 
came applicable to all numbers of the first series. Finally, when 
the still more general symbol, L -\- H, was substituted for the 3 
of the earlier formula, the associated meanings became appli- 
cable to all problems of all continuous series. This is in accord 
with Judd's view that application is much facilitated by gener- 
alization.^^ 

It is of course possible for knowledge to be expressed in gen- 
eral form without necessarily being generalized. There is, as 
Dewey observes, "always danger that symbols will not be truly 
representative; danger that instead of calling up the absent and 
remote in the way to make it enter a present experience, the lin- 
guistic media of representation will become an end in them- 

* This tendency was probably strengthened in our experiments by the gen- 
eral conception of the problem. 

Speaking of the influence of specific associations upon thinking, Thorn- 
dike says : "It has long been apparent that man's erroneous inferences — his 
vinsuccessful responses to novel situations — are due to the action of mis- 
leading connections and analogies to which he is led by the laws of habit. 
It is also the fact, though it is not so apparent, that his successful responses 
are due to fruitful connections and analogies to which he is led by the same 
laws." (Educational Psychology, vol. 11, p. 48.) The operation of the law 
of habit is here, however, as obvious in the arousal of correct as of incor- 
rect responses through the medium of the general symbol. 

20 Bode argues that "transfer of training, when translated into terms of 
presentday knowledge, means the extension or applicaiton of meanings to 
new problems or new situations." (Boyd Henry Bode, "A Reinterpretation 
of Transfer of Training," Educational Administration and Supervision, vol. 
V, 1919, p. 107.) 

2ijudd, C. H. "Psychology of High-School Subjects," pp. 392-435. 



HIGHER MENTAL PROCESSES IN LEARNING 119 

selves.""" Even though the symbol may previously have acquired 
a wide range of meaningful associations, it may, through the 
subject's lack of acquaintance with the elements of the present 
situation, function as a mere verbal response. Thus the hypothe- 
sis that that all multiples of 3 are insoluble often failed to func- 
tion in any noticeable degree until long after its first explicit 
formulation, if such formulation chanced to occur before a fair 
acquaintance with the elements of the situation had been gained. 
That the degree of acquaintance with the elements strongly af- 
fects the applicability of generalizations to new problems, is also 
suggested by the fact that the degree of transfer from Series 1-2 
to Series 1-3 was much greater for subjects whose progress 
through the first series was slow than for those who progressed 
rapidly. 

The discovery of new elements of the situation through the 
associative action of a general symbol and the transfer of old 
meanings to these elements through the same medium, must in- 
evitably lead to errors, since there is no certainty that the re- 
quired symbol will be the first aroused or that when once aroused 
this symbol will call up only the correct association. Trial and 
error is obviously an essential method of procedure so long as 
any of the elements of the old or the new situation remain un- 
known, i.e., whenever new applications are being attempted. If 
the old situation is not fully analyzed, the generalization may be 
erroneous or inadequate; if analysis of the new situation is not 
complete, nonessential elements may dominate attention and thus 
bring about the arousal of inappropriate conceptions. 

From the foregoing facts it is apparent that transfer is much 
facihtated by explicit generalization of experiences. This facili- 
tation can be accounted for on the basis of the operation of 
specific associations between the symbols required for generali- 
zation and the elements of direct experience, which furnish the 
materials for generalization. The degree of transfer appears to 
depend upon both the number and the strength of such associa- 
tions. It is also dependent largely upon the degree of acquaint- 
ance with the elements of the problematic situation to which ap- 

22 Dewey, J. "Democracy and Education," p. 272. 



120 JOHN C. PETERSON 

plication is to be affected. The educational value of a habit of 
analyzing and generalizing experiences, which may, according to 
Judd,-^ be developed by constant exercise of those functions, is 
beyond question. But it should not be forgotten that this habit 
must function in more or less familiar materials and through the 
operation of specific associations. Its value depends far more 
upon a gradual ordering of daily experiences, through repeated 
attempts at analysis and generalization, into such a flexible, sys- 
tematic form as will make them available for future use, than 
upon sudden transformation of experiences at the moment of 
need. 

D. General Conclusions. — Trial and error appears to be a uni- 
versal method of procedure in learning of the problem-solving 
type. Not only does this procedure dominate the early stages of 
analysis of new materials, but it is a conspicuous factor in the 
determination of progress in the generalization of knowledge and 
its application to new situations. The field of variation is gradu- 
ally limited through the effects of generalization, and the testing 
out of trial responses is facilitated through the familiarization 
of the elements of the situation. Thus larger and larger units 
of response come to be represented ideationally and tested out 
either overtly or in imagination ; but the general trial-and-error 
character of the process remains always the same. 

The most obvious factors in the selection and accentuation of 
essential elements were frequency of repetition of elements and 
their relative nearness to a goal, or end of action. Generalization 
and application of experiences were apparently somewhat less, 
though still largely, controlled by the same factors. 

Progress in the detachment of elements and their generaliza- 
tion and application to new situations can be traced largely to 
the gradual formation and automatization of specific associations 
and to the associative arousal of previously formed concepts. 
But it should not be forgotten that old concepts wrung into 
service in this manner are themselves the product of earlier pro- 
cesses of gradual formation and mechanization of associations, 
essentially similar to the learning of sensori-motor co-ordina- 

23 Op. cit., pp. 432-435- 



HIGHER MENTAL PROCESSES IN LEARNING 121 

tions. The learning process is not so much modified as abridged 
by this action of old concepts. The difference between sensori- 
motor learning and learning through abstraction and generaliza- 
tion is not so much a difference of method as of the type and 
complexity of previously established inter-relations of the ma- 
terials to be organized. All of our data appear to confirm the 
view recently expressed by Thorndike that "Thinking and reas- 
oning do not seem in any useful sense opposites of automatism, 
custom, or habit, but simply the action of habits in cases where 
the elements of the situation compete and co-operate notably."'^* 
• All ideas of theoretical importance which are expressed in the 
foregoing pages have been stated previously by other writers. 
The only originality which is claimed for the present study is 
to be found in the experimental verification of some of these 
ideas and in the development of a technique which, it is hoped, 
may prove to be of value in the further investigation of some of 
the problems in this field. 

2* Thorndike, E. L. "The Psychology of Thinking in the Case of Read- 
ing," Psychological Review, vol. XXIV, 1917, pp. 233-234. 



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